diff --git a/core/src/main/scala/cats/data/Inverse.scala b/core/src/main/scala/cats/data/Inverse.scala
new file mode 100644
index 0000000000..00a4c8beca
--- /dev/null
+++ b/core/src/main/scala/cats/data/Inverse.scala
@@ -0,0 +1,123 @@
+/*
+ * Copyright (c) 2015 Typelevel
+ *
+ * Permission is hereby granted, free of charge, to any person obtaining a copy of
+ * this software and associated documentation files (the "Software"), to deal in
+ * the Software without restriction, including without limitation the rights to
+ * use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
+ * the Software, and to permit persons to whom the Software is furnished to do so,
+ * subject to the following conditions:
+ *
+ * The above copyright notice and this permission notice shall be included in all
+ * copies or substantial portions of the Software.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+ * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS
+ * FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR
+ * COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER
+ * IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+ * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+ */
+
+package cats
+package data
+
+import cats.kernel.LowerBounded
+import cats.kernel.UpperBounded
+
+object InverseImpl extends InverseInstances with Newtype {
+  private[data] def unwrap[A](value: Type[A]): A =
+    value.asInstanceOf[A]
+
+  private[data] def create[A](value: A): Type[A] =
+    value.asInstanceOf[Type[A]]
+
+  def apply[A](value: A): Inverse[A] =
+    create(value)
+
+  implicit def catsInverseOps[A](value: Inverse[A]): InverseOps[A] =
+    new InverseOps[A](value)
+}
+
+sealed class InverseOps[A](val inverse: Inverse[A]) {
+  def value: A = InverseImpl.unwrap(inverse)
+}
+
+sealed abstract private[data] class InverseInstances extends InverseInstances0 {
+  implicit def hashAndOrderForInverse[A: Hash: Order]: Hash[Inverse[A]] with Order[Inverse[A]] =
+    new Hash[Inverse[A]] with Order[Inverse[A]] {
+      override def hash(x: Inverse[A]): Int =
+        Hash[A].hash(x.value)
+
+      override def compare(x: Inverse[A], y: Inverse[A]): Int =
+        Order.reverse[A](Order[A]).compare(x.value, y.value)
+    }
+
+  implicit def orderingForInverseFromHashAndOrder[A](implicit A: Order[Inverse[A]]): Ordering[Inverse[A]] =
+    A.toOrdering
+
+  implicit def lowerBoundForInverse[A](implicit
+    A: UpperBounded[A],
+    B: PartialOrder[Inverse[A]]
+  ): LowerBounded[Inverse[A]] =
+    new LowerBounded[Inverse[A]] {
+      override def partialOrder: PartialOrder[Inverse[A]] = B
+
+      override def minBound: Inverse[A] =
+        Inverse(A.maxBound)
+    }
+
+  implicit def upperBoundForInverse[A](implicit
+    A: LowerBounded[A],
+    B: PartialOrder[Inverse[A]]
+  ): UpperBounded[Inverse[A]] =
+    new UpperBounded[Inverse[A]] {
+      override def partialOrder: PartialOrder[Inverse[A]] = B
+
+      override def maxBound: Inverse[A] =
+        Inverse(A.minBound)
+    }
+}
+
+sealed private[data] trait InverseInstances0 extends InverseInstances1 {
+  implicit def hashAndPartialOrderForInverse[A: Hash: PartialOrder]: Hash[Inverse[A]] with PartialOrder[Inverse[A]] =
+    new Hash[Inverse[A]] with PartialOrder[Inverse[A]] {
+      override def hash(x: Inverse[A]): Int =
+        Hash[A].hash(x.value)
+
+      override def partialCompare(x: Inverse[A], y: Inverse[A]): Double =
+        PartialOrder.reverse(PartialOrder[A]).partialCompare(x.value, y.value)
+    }
+}
+
+sealed private[data] trait InverseInstances1 extends InverseInstances2 {
+  implicit def orderForInverse[A: Order]: Order[Inverse[A]] =
+    new Order[Inverse[A]] {
+      override def compare(x: Inverse[A], y: Inverse[A]): Int =
+        Order.reverse(Order[A]).compare(x.value, y.value)
+    }
+}
+
+sealed private[data] trait InverseInstances2 extends InverseInstances3 {
+  implicit def partialOrderForInverse[A: PartialOrder]: PartialOrder[Inverse[A]] =
+    new PartialOrder[Inverse[A]] {
+      override def partialCompare(x: Inverse[A], y: Inverse[A]): Double =
+        PartialOrder.reverse(PartialOrder[A]).partialCompare(x.value, y.value)
+    }
+}
+
+sealed private[data] trait InverseInstances3 extends InverseInstances4 {
+  implicit def hashForInverse[A: Hash]: Hash[Inverse[A]] =
+    new Hash[Inverse[A]] {
+      override def hash(x: Inverse[A]): Int =
+        Hash[A].hash(x.value)
+
+      override def eqv(x: Inverse[A], y: Inverse[A]): Boolean =
+        Hash[A].eqv(x.value, y.value)
+    }
+}
+
+sealed private[data] trait InverseInstances4 {
+  implicit def eqForInverse[A: Eq]: Eq[Inverse[A]] =
+    Eq.by[Inverse[A], A](_.value)
+}
diff --git a/core/src/main/scala/cats/data/package.scala b/core/src/main/scala/cats/data/package.scala
index 4ac5961361..38e61342b2 100644
--- a/core/src/main/scala/cats/data/package.scala
+++ b/core/src/main/scala/cats/data/package.scala
@@ -41,6 +41,29 @@ package object data extends ScalaVersionSpecificPackage {
   type NonEmptyChain[+A] = NonEmptyChainImpl.Type[A]
   val NonEmptyChain = NonEmptyChainImpl
 
+  /** Inverse is a data type which exists merely to invert the `Order`,
+    * `PartialOrder`, `UpperBounded`, and `LowerBounded` instances for some
+    * type `A`. In some languages this type is called `Down`.
+    *
+    * {{{
+    * scala> import cats.data._
+    * import cats.data._
+    *
+    * scala> import scala.collection.immutable.SortedSet
+    * import scala.collection.immutable.SortedSet
+    *
+    * scala> SortedSet(1, 2, 3)
+    * val res0: scala.collection.immutable.SortedSet[Int] = TreeSet(1, 2, 3)
+    *
+    * scala> res0.map(Inverse.apply)
+    * val res1: scala.collection.immutable.SortedSet[cats.data.Inverse[Int]] = TreeSet(3, 2, 1)
+    * }}}
+    *
+    * @see [[https://hackage.haskell.org/package/base-4.6.0.1/docs/Data-Ord.html#Down]]
+    */
+  type Inverse[A] = InverseImpl.Type[A]
+  val Inverse = InverseImpl
+
   type ReaderT[F[_], -A, B] = Kleisli[F, A, B]
   val ReaderT = Kleisli