diff --git a/astropy_healpix/representation.py b/astropy_healpix/representation.py
new file mode 100644
index 0000000..ca9200e
--- /dev/null
+++ b/astropy_healpix/representation.py
@@ -0,0 +1,178 @@
+import astropy_healpix
+from abc import abstractmethod
+from astropy.coordinates.representation import (
+    REPRESENTATION_CLASSES,
+    DUPLICATE_REPRESENTATIONS,
+    BaseRepresentation,
+    CartesianRepresentation,
+)
+from astropy.coordinates import Distance
+import astropy.units as u
+
+
+class HEALPixRepresentationBase(BaseRepresentation):
+
+    attr_classes = {
+        "indices": u.Quantity,
+        "dx": u.Quantity,
+        "dy": u.Quantity,
+    }
+
+    _order: int
+    _nside: int
+    _dx: u.Quantity
+    _dy: u.Quantity
+
+    @classmethod
+    def get_subclass(cls, *, order, nside):
+        name: str = f"HEALPix{order.capitalize()}{nside}"
+        if "Unit" in cls.__name__:
+            name += "Unit"
+
+        if name.lower() in REPRESENTATION_CLASSES:
+            return REPRESENTATION_CLASSES[name.lower()]
+        elif "abc." + name in REPRESENTATION_CLASSES:
+            return REPRESENTATION_CLASSES["abc." + name]
+
+        return type(name + "Representation", (cls,), {}, order=order, nside=nside)
+
+    @classmethod
+    @property
+    @abstractmethod
+    def order(cls):
+        return cls._order
+
+    @classmethod
+    @property
+    @abstractmethod
+    def nside(cls):
+        return cls._nside
+
+    def __init__(self, *args, differentials=None, copy=True):
+        super().__init__(*args, differentials=differentials, copy=copy)
+        self._indices = (self._indices << u.one).astype(int)
+        self._dx <<= u.one
+        self._dy <<= u.one
+
+    @property
+    def indices(self):
+        """HEALPix indices."""
+        return self._indices
+
+    @property
+    def dx(self):
+        """HEALPix x offset."""
+        return self._dx
+
+    @property
+    def dy(self):
+        """HEALPix y offset."""
+        return self._dy
+
+
+class HEALPixBaseRepresentation(HEALPixRepresentationBase):
+
+    attr_classes = {
+        "indices": u.Quantity,
+        "distance": Distance,
+        "dx": u.Quantity,
+        "dy": u.Quantity,
+    }
+
+    _distance: Distance
+
+    def __init_subclass__(cls, *, order: str, nside: int, **kwargs):
+        super().__init_subclass__(**kwargs)
+
+        astropy_healpix.core._validate_nside(nside)
+        cls._nside = int(nside)
+
+        astropy_healpix.core._validate_order(order)
+        cls._order = order
+
+    @classmethod
+    @property
+    @abstractmethod
+    def order(cls):
+        return cls._order
+
+    @classmethod
+    @property
+    @abstractmethod
+    def nside(cls):
+        return cls._nside
+
+    def __init__(
+        self, indices, distance, dx=0.5, dy=0.5, differentials=None, copy=True
+    ):
+        super().__init__(
+            indices, distance, dx, dy, differentials=differentials, copy=copy
+        )
+
+    @property
+    def distance(self):
+        """The distance from the origin to the point(s)."""
+        return self._distance
+
+    def to_cartesian(self):
+        # TODO! register astropy_healpix.healpix_to_xyz with Quantity
+        xyz = astropy_healpix.healpix_to_xyz(
+            self.indices.value,
+            self.nside,
+            dx=self.dx.value,
+            dy=self.dy.value,
+            order=self.order,
+        )
+        return CartesianRepresentation(*xyz * self.distance)
+
+    @classmethod
+    def from_cartesian(cls, cart):
+        distance = cart.norm()
+        indices, dx, dy = astropy_healpix.xyz_to_healpix(
+            *cart.xyz.value, nside=cls.nside, return_offsets=True, order=cls.order
+        )
+        return cls(indices, distance, dx=dx, dy=dy)
+
+
+class HEALPixBaseUnitRepresentation(HEALPixRepresentationBase):
+    def __init_subclass__(cls, *, order: str, nside: int, **kwargs):
+        super().__init_subclass__(**kwargs)
+
+        astropy_healpix.core._validate_nside(nside)
+        cls._nside = int(nside)
+
+        astropy_healpix.core._validate_order(order)
+        cls._order = order
+
+    def __init__(self, indices, dx=0.5, dy=0.5, differentials=None, copy=True):
+        super().__init__(indices, dx, dy, differentials=differentials, copy=copy)
+
+    @classmethod
+    @property
+    @abstractmethod
+    def order(cls):
+        return cls._order
+
+    @classmethod
+    @property
+    @abstractmethod
+    def nside(cls):
+        return cls._nside
+
+    def to_cartesian(self):
+        xyz = astropy_healpix.healpix_to_xyz(
+            self.indices.value,
+            self.nside,
+            dx=self.dx.value,
+            dy=self.dy.value,
+            order=self.order,
+        )
+        return CartesianRepresentation(*xyz)
+
+    @classmethod
+    def from_cartesian(cls, cart):
+        distance = cart.norm()
+        indices, dx, dy = astropy_healpix.xyz_to_healpix(
+            *cart.xyz.value, nside=cls.nside, return_offsets=True, order=cls.order
+        )
+        return cls(indices, dx=dx, dy=dy)