diff --git a/curvlinops/_torch_base.py b/curvlinops/_torch_base.py
index 4937bfc..97c4aaf 100644
--- a/curvlinops/_torch_base.py
+++ b/curvlinops/_torch_base.py
@@ -32,8 +32,14 @@ class PyTorchLinearOperator:
     operator, which can be useful for interfacing with SciPy routines. To achieve this,
     the functions ``_infer_device`` and ``_infer_dtype`` must be implemented.
 
+    Attributes:
+        SELF_ADJOINT: Whether the linear operator is self-adjoint. If ``True``,
+            ``_adjoint`` does not need to be implemented. Default: ``False``.
+
     """
 
+    SELF_ADJOINT: bool = False
+
     def __init__(
         self, in_shape: List[Tuple[int, ...]], out_shape: List[Tuple[int, ...]]
     ):
@@ -106,7 +112,7 @@ def adjoint(self) -> PyTorchLinearOperator:
         Returns:
             The adjoint of the linear operator.
         """
-        return self._adjoint()
+        return self if self.SELF_ADJOINT else self._adjoint()
 
     def _adjoint(self) -> PyTorchLinearOperator:
         """Adjoint of the linear operator.
diff --git a/curvlinops/hessian.py b/curvlinops/hessian.py
index e64abdd..2bae30b 100644
--- a/curvlinops/hessian.py
+++ b/curvlinops/hessian.py
@@ -1,7 +1,5 @@
 """Contains a linear operator implementation of the Hessian."""
 
-from __future__ import annotations
-
 from collections.abc import MutableMapping
 from typing import List, Union
 
@@ -37,8 +35,11 @@ class HessianLinearOperator(CurvatureLinearOperator):
     Attributes:
         SUPPORTS_BLOCKS: Whether the linear operator supports block operations.
             Default is ``True``.
+        SELF_ADJOINT: Whether the linear operator is self-adjoint (``True`` for
+            Hessians).
     """
 
+    SELF_ADJOINT: bool = True
     SUPPORTS_BLOCKS: bool = True
 
     def _matmat_batch(
@@ -81,13 +82,3 @@ def _matmat_batch(
                     AM_block[p][..., n].add_(col)
 
         return AM
-
-    def _adjoint(self) -> HessianLinearOperator:
-        """Return the linear operator representing the adjoint.
-
-        The Hessian is real symmetric, and hence self-adjoint.
-
-        Returns:
-            Self.
-        """
-        return self