diff --git a/README.md b/README.md
index 9a27d0c..db44957 100644
--- a/README.md
+++ b/README.md
@@ -168,7 +168,14 @@ bool isGreater = number1 > number2;
 bool isLessOrEqual = number1 <= number2;
 ```
 
-Ordering operators compare values exactly. `SignificantNumber.CompareTo(left, right)` and `CompareTo(SignificantNumber)` compare both numbers at the lower of their significant digit counts.
+Ordering operators, `Equals` and `CompareTo` compare values exactly, so `Array.Sort`, `OrderBy`, `SortedSet` and `Comparer<SignificantNumber>.Default` all see one consistent order.
+
+To ask whether two measurements agree to the precision they share, use `SignificantNumber.CompareAtSignificance(left, right)`. It compares both numbers at the lower of their significant digit counts, so `1.23` and `1.2` compare equal. That comparison is not transitive (`1.2` also equals `1.17`, but `1.23` does not), so don't sort with it.
+
+```csharp
+int exact = number1.CompareTo(number2);
+int atSharedPrecision = SignificantNumber.CompareAtSignificance(number1, number2);
+```
 
 ### Formatting and parsing
 
@@ -306,7 +313,7 @@ Consider the number `123.456000`:
 
 - `bool Equals(SignificantNumber other)` - Determines whether two numbers have the same significand and exponent.
 - `int CompareTo(object? obj)` - Compares the current instance with another object.
-- `int CompareTo(SignificantNumber other)` - Compares the current instance with another significant number at the lower of their significant digit counts.
+- `int CompareTo(SignificantNumber other)` - Compares the value of the current instance with another significant number exactly.
 - `int CompareTo<TInput>(TInput other) where TInput : INumber<TInput>` - Compares the value of the current instance with another number.
 - `PreciseNumber Abs()` - Returns the absolute value of the current instance.
 - `PreciseNumber Round(int decimalDigits)` - Rounds the current instance to the specified number of decimal digits.
@@ -324,6 +331,7 @@ Consider the number `123.456000`:
 - `static SignificantNumber Add`, `Subtract`, `Multiply`, `Divide`, and `Mod(PreciseNumber left, PreciseNumber right)` - Apply the significant figure rules to two numbers.
 - `static SignificantNumber Exp(PreciseNumber power)` - Raises e to a power.
 - `static SignificantNumber Max`, `Min(SignificantNumber x, SignificantNumber y)`, and `Clamp(SignificantNumber value, SignificantNumber min, SignificantNumber max)` - Compare by value.
+- `static int CompareAtSignificance(PreciseNumber left, PreciseNumber right)` - Compares two numbers at the lower of their significant digit counts. Not a total order, so not for sorting.
 - `static SignificantNumber Round(SignificantNumber value, int decimalDigits)` - Rounds a number to the specified number of decimal digits.
 - `static SignificantNumber Abs(SignificantNumber value)` - Returns the absolute value of a `SignificantNumber`.
 - `static bool IsCanonical(SignificantNumber value)` - Determines whether the specified value is canonical.
diff --git a/SignificantNumber.Test/SignificantNumberTests.cs b/SignificantNumber.Test/SignificantNumberTests.cs
index 4cebeee..f21ed8d 100644
--- a/SignificantNumber.Test/SignificantNumberTests.cs
+++ b/SignificantNumber.Test/SignificantNumberTests.cs
@@ -178,13 +178,13 @@ public void NotEqual_TwoNumbers_ReturnsTrue()
 	}
 
 	[TestMethod]
-	public void CompareTo_TwoNumbers_ReturnsCorrectComparison()
+	public void CompareAtSignificance_TwoNumbers_ReturnsCorrectComparison()
 	{
 		SignificantNumber left = SignificantNumber.CreateFromComponents(1, new BigInteger(10));
 		SignificantNumber right = SignificantNumber.CreateFromComponents(1, new BigInteger(5));
-		int result = SignificantNumber.CompareTo(left, right);
+		int result = SignificantNumber.CompareAtSignificance(left, right);
 
-		Assert.IsGreaterThan(0, result, "CompareTo should return a positive value when left (10) is greater than right (5)");
+		Assert.IsGreaterThan(0, result, "CompareAtSignificance should return a positive value when left (10) is greater than right (5)");
 	}
 
 	[TestMethod]
diff --git a/SignificantNumber.Test/SignificantNumberValueTypeTests.cs b/SignificantNumber.Test/SignificantNumberValueTypeTests.cs
index 65ea469..c4a6b00 100644
--- a/SignificantNumber.Test/SignificantNumberValueTypeTests.cs
+++ b/SignificantNumber.Test/SignificantNumberValueTypeTests.cs
@@ -231,4 +231,68 @@ public void ToSignificantNumberWithDigits_FromSignificantNumber_ReducesSignifica
 
 		Assert.AreEqual(Parse("123"), result);
 	}
+
+	// IComparable is what Array.Sort, OrderBy, SortedSet and Comparer<T>.Default use, so it has to be a
+	// consistent total order. It used to compare at the lower of the two significant digit counts, under
+	// which 1.23 == 1.2 and 1.2 == 1.17 but 1.23 > 1.17, so sorting gave a wrong order.
+
+	[TestMethod]
+	public void ArraySort_OrdersByExactValue()
+	{
+		SignificantNumber[] values = [Parse("1.17"), Parse("1.23"), Parse("1.2"), Parse("1.21"), Parse("1.19")];
+		SignificantNumber[] expected = [Parse("1.17"), Parse("1.19"), Parse("1.2"), Parse("1.21"), Parse("1.23")];
+
+		Array.Sort(values);
+
+		Assert.AreSequenceEqual(expected, values);
+		Assert.AreSequenceEqual(expected, values.Reverse().OrderBy(x => x));
+	}
+
+	[TestMethod]
+	public void SortedSet_KeepsValuesThatDifferOnlyBelowTheSharedSignificance()
+	{
+		SortedSet<SignificantNumber> set = [Parse("1.23"), Parse("1.2"), Parse("1.17")];
+
+		Assert.HasCount(3, set);
+		Assert.Contains(Parse("1.2"), set);
+	}
+
+	[TestMethod]
+	public void CompareTo_AgreesWithEqualityAndOrderingOperators()
+	{
+		SignificantNumber[] values = [Parse("1.17"), Parse("1.19"), Parse("1.2"), Parse("1.21"), Parse("1.23"), Parse("1.20"), Parse("-3"), Parse("0"), Parse("1200")];
+
+		foreach (SignificantNumber left in values)
+		{
+			foreach (SignificantNumber right in values)
+			{
+				int comparison = left.CompareTo(right);
+				Assert.AreEqual(left == right, comparison == 0, $"{left}.CompareTo({right}) == 0");
+				Assert.AreEqual(left < right, comparison < 0, $"{left}.CompareTo({right}) < 0");
+				Assert.AreEqual(left > right, comparison > 0, $"{left}.CompareTo({right}) > 0");
+				Assert.AreEqual(int.Sign(comparison), int.Sign(left.CompareTo((object)right)), $"{left}.CompareTo((object){right})");
+				Assert.AreEqual(int.Sign(comparison), int.Sign(left.CompareTo(right.Value)), $"{left}.CompareTo({right}.Value)");
+			}
+		}
+	}
+
+	[TestMethod]
+	public void CompareToObject_BoxedPreciseNumber_ComparesExactValue()
+	{
+		object precise = PreciseNumber.Parse("1.2", CultureInfo.InvariantCulture);
+
+		Assert.IsGreaterThan(0, Parse("1.23").CompareTo(precise));
+	}
+
+	[TestMethod]
+	[DataRow("1.23", "1.2", 0)]
+	[DataRow("1.2", "1.17", 0)]
+	[DataRow("1.23", "1.17", 1)]
+	[DataRow("1.17", "1.23", -1)]
+	[DataRow("1200", "1234", 0)]
+	[DataRow("1.5", "1.4", 1)]
+	public void CompareAtSignificance_ComparesAtTheLowerSignificantDigitCount(string left, string right, int expected)
+	{
+		Assert.AreEqual(expected, int.Sign(SignificantNumber.CompareAtSignificance(Parse(left), Parse(right))));
+	}
 }
diff --git a/SignificantNumber/SignificantNumber.cs b/SignificantNumber/SignificantNumber.cs
index 9191bf9..c90ff9e 100644
--- a/SignificantNumber/SignificantNumber.cs
+++ b/SignificantNumber/SignificantNumber.cs
@@ -364,6 +364,11 @@ public static bool NotEqual(PreciseNumber left, PreciseNumber right) =>
 	/// <summary>
 	/// Compares two numbers at the lower of their significant digit counts.
 	/// </summary>
+	/// <remarks>
+	/// This is not a total order: <c>1.23</c> and <c>1.2</c> compare equal, as do <c>1.2</c> and <c>1.17</c>, but
+	/// <c>1.23</c> is greater than <c>1.17</c>. Use it to ask whether two measurements agree to the precision they
+	/// share, and use <see cref="CompareTo(SignificantNumber)"/> to sort.
+	/// </remarks>
 	/// <param name="left">The first number to compare.</param>
 	/// <param name="right">The second number to compare.</param>
 	/// <returns>
@@ -380,7 +385,7 @@ public static bool NotEqual(PreciseNumber left, PreciseNumber right) =>
 	/// </item>
 	/// </list>
 	/// </returns>
-	public static int CompareTo(PreciseNumber left, PreciseNumber right)
+	public static int CompareAtSignificance(PreciseNumber left, PreciseNumber right)
 	{
 		int lowestSignificantDigits = LowestSignificantDigits(left, right);
 		return left.ReduceSignificance(lowestSignificantDigits).CompareTo(right.ReduceSignificance(lowestSignificantDigits));
@@ -787,7 +792,7 @@ private static PreciseNumber ExactIntegerPower(PreciseNumber value, BigInteger e
 	}
 
 	/// <summary>
-	/// Compares the current instance with another <see cref="SignificantNumber"/> at the lower of their significant digit counts.
+	/// Compares the value of the current instance with another <see cref="SignificantNumber"/>.
 	/// </summary>
 	/// <param name="other">The <see cref="SignificantNumber"/> to compare with the current instance.</param>
 	/// <returns>
@@ -804,8 +809,14 @@ private static PreciseNumber ExactIntegerPower(PreciseNumber value, BigInteger e
 	/// </item>
 	/// </list>
 	/// </returns>
+	/// <remarks>
+	/// The values are compared exactly, without applying significant figure rules, so the order agrees with
+	/// <c>==</c>, <c>&lt;</c> and <c>&gt;</c> and is a consistent total order for sorting. Use
+	/// <see cref="CompareAtSignificance(PreciseNumber, PreciseNumber)"/> to compare at the lower of the two
+	/// significant digit counts.
+	/// </remarks>
 	public int CompareTo(SignificantNumber other) =>
-		CompareTo(this, other);
+		Value.CompareTo(other.Value);
 
 	/// <summary>
 	/// Compares the current instance with an object.
@@ -816,15 +827,15 @@ public int CompareTo(SignificantNumber other) =>
 	/// <see langword="null"/> object sorts before every number.
 	/// </returns>
 	/// <remarks>
-	/// A <see cref="SignificantNumber"/> or <see cref="PreciseNumber"/> is compared at the lower of the two significant
-	/// digit counts. Any other object is passed to <see cref="PreciseNumber.CompareTo(object)"/>.
+	/// A <see cref="SignificantNumber"/> or <see cref="PreciseNumber"/> is compared by exact value, the same as
+	/// <see cref="CompareTo(SignificantNumber)"/>. Any other object is passed to <see cref="PreciseNumber.CompareTo(object)"/>.
 	/// </remarks>
 	public int CompareTo(object? obj) =>
 		obj switch
 		{
 			null => 1,
-			SignificantNumber significantNumber => CompareTo(significantNumber),
-			PreciseNumber preciseNumber => CompareTo(this, preciseNumber),
+			SignificantNumber significantNumber => Value.CompareTo(significantNumber.Value),
+			PreciseNumber preciseNumber => Value.CompareTo(preciseNumber),
 			_ => Value.CompareTo(obj),
 		};