diff --git a/crates/fj-kernel/src/objects/full/edge.rs b/crates/fj-kernel/src/objects/full/edge.rs
index 65216649b..893aaef70 100644
--- a/crates/fj-kernel/src/objects/full/edge.rs
+++ b/crates/fj-kernel/src/objects/full/edge.rs
@@ -105,6 +105,20 @@ impl HalfEdge {
 ///
 /// See [`HalfEdge`]'s documentation for more information on the relationship
 /// between [`HalfEdge`] and `GlobalEdge`.
+///
+/// # Equality
+///
+/// `GlobalEdge` contains no data and exists purely to be used within a
+/// `Handle`, where `Handle::id` can be used to compare different instances of
+/// `GlobalEdge`.
+///
+/// If `GlobalEdge` had `Eq`/`PartialEq` implementations, it containing no data
+/// would mean that all instances of `GlobalEdge` would be considered equal.
+/// This would be very error-prone.
+///
+/// If you need to reference a `GlobalEdge` from a struct that needs to derive
+/// `Eq`/`Ord`/..., you can use `HandleWrapper<GlobalEdge>` to do that. It will
+/// use `Handle::id` to provide those `Eq`/`Ord`/... implementations.
 #[derive(Clone, Debug, Default, Hash)]
 pub struct GlobalEdge {}
 
diff --git a/crates/fj-kernel/src/objects/full/vertex.rs b/crates/fj-kernel/src/objects/full/vertex.rs
index ee44ab0bf..10d5a80f6 100644
--- a/crates/fj-kernel/src/objects/full/vertex.rs
+++ b/crates/fj-kernel/src/objects/full/vertex.rs
@@ -15,6 +15,19 @@
 /// between distinct vertices can be configured using the respective field in
 /// [`ValidationConfig`].
 ///
+/// # Equality
+///
+/// `Vertex` contains no data and exists purely to be used within a `Handle`,
+/// where `Handle::id` can be used to compare different instances of `Vertex`.
+///
+/// If `Vertex` had `Eq`/`PartialEq` implementations, it containing no data
+/// would mean that all instances of `Vertex` would be considered equal. This
+/// would be very error-prone.
+///
+/// If you need to reference a `Vertex` from a struct that needs to derive
+/// `Eq`/`Ord`/..., you can use `HandleWrapper<Vertex>` to do that. It will
+/// use `Handle::id` to provide those `Eq`/`Ord`/... implementations.
+///
 /// [`ValidationConfig`]: crate::validate::ValidationConfig
 #[derive(Clone, Copy, Debug, Default, Hash)]
 pub struct Vertex {}
diff --git a/crates/fj-kernel/src/objects/object.rs b/crates/fj-kernel/src/objects/object.rs
index 2c22aa16f..22c9d16f2 100644
--- a/crates/fj-kernel/src/objects/object.rs
+++ b/crates/fj-kernel/src/objects/object.rs
@@ -3,7 +3,7 @@ use crate::{
         Cycle, Face, GlobalEdge, HalfEdge, Objects, Shell, Sketch, Solid,
         Surface, Vertex,
     },
-    storage::{Handle, ObjectId},
+    storage::{Handle, HandleWrapper, ObjectId},
     validate::{Validate, ValidationError},
 };
 
@@ -14,7 +14,7 @@ macro_rules! object {
         /// This enum is generic over the form that the object takes. An
         /// `Object<Bare>` contains bare objects, like `Curve`. An
         /// `Object<BehindHandle>` contains handles, like `Handle<Curve>`.
-        #[derive(Clone, Debug)]
+        #[derive(Clone, Debug, Eq, PartialEq, Ord, PartialOrd)]
         pub enum Object<F: Form> {
             $(
                 #[doc = concat!("A ", $name)]
@@ -39,8 +39,10 @@ macro_rules! object {
                 match self {
                     $(
                         Self::$ty((handle, object)) => {
-                            objects.$store.insert(handle.clone(), object);
-                            handle.into()
+                            objects.$store.insert(
+                                handle.clone().into(), object
+                            );
+                            handle.0.into()
                         }
                     )*
                 }
@@ -60,7 +62,7 @@ macro_rules! object {
             fn from(object: Object<WithHandle>) -> Self {
                 match object {
                     $(
-                        Object::$ty((handle, _)) => Self::$ty(handle),
+                        Object::$ty((handle, _)) => Self::$ty(handle.into()),
                     )*
                 }
             }
@@ -75,13 +77,13 @@ macro_rules! object {
 
             impl From<Handle<$ty>> for Object<BehindHandle> {
                 fn from(object: Handle<$ty>) -> Self {
-                    Self::$ty(object)
+                    Self::$ty(object.into())
                 }
             }
 
             impl From<(Handle<$ty>, $ty)> for Object<WithHandle> {
                 fn from((handle, object): (Handle<$ty>, $ty)) -> Self {
-                    Self::$ty((handle, object))
+                    Self::$ty((handle.into(), object))
                 }
             }
         )*
@@ -110,7 +112,7 @@ pub trait Form {
 }
 
 /// Implementation of [`Form`] for bare objects
-#[derive(Clone, Debug)]
+#[derive(Clone, Debug, Eq, PartialEq, Ord, PartialOrd)]
 pub struct Bare;
 
 impl Form for Bare {
@@ -118,17 +120,17 @@ impl Form for Bare {
 }
 
 /// Implementation of [`Form`] for objects behind a handle
-#[derive(Clone, Debug)]
+#[derive(Clone, Debug, Eq, PartialEq, Ord, PartialOrd)]
 pub struct BehindHandle;
 
 impl Form for BehindHandle {
-    type Form<T> = Handle<T>;
+    type Form<T> = HandleWrapper<T>;
 }
 
 /// Implementation of [`Form`] for objects that are paired with their handle
-#[derive(Clone, Debug)]
+#[derive(Clone, Debug, Eq, PartialEq, Ord, PartialOrd)]
 pub struct WithHandle;
 
 impl Form for WithHandle {
-    type Form<T> = (Handle<T>, T);
+    type Form<T> = (HandleWrapper<T>, T);
 }