diff --git a/scrise/factorization.py b/scrise/factorization.py
index ea17061f..3ae2c582 100644
--- a/scrise/factorization.py
+++ b/scrise/factorization.py
@@ -121,8 +121,12 @@ def order_components_by_energy(X: anndata.AnnData) -> anndata.AnnData:
     -------
     anndata.AnnData
         The same AnnData object, with Pf2_A, Pf2_B, Pf2_C, Pf2_weights
-        (and, if present, obsm["weighted_projections"]) reordered/updated
-        in place.
+        (and, if present, obsm["projections"] and
+        obsm["weighted_projections"]) reordered/updated in place. The
+        eigen-state axis of Pf2_B, and the matching columns of
+        obsm["projections"], are permuted alongside the components so that
+        the maximal-diagonal form of B established by
+        ``parafac2.utils.standardize_pf2`` is preserved.
     """
     A = np.array(X.uns["Pf2_A"])
     B = np.array(X.uns["Pf2_B"])
@@ -141,10 +145,19 @@ def order_components_by_energy(X: anndata.AnnData) -> anndata.AnnData:
 
     X.uns["Pf2_A"] = A[:, order]
     X.varm["Pf2_C"] = C[:, order]
-    X.uns["Pf2_B"] = B[:, order]
     X.uns["Pf2_weights"] = weights[order]
 
+    # B is indexed by (eigen-state, component). ``parafac2.utils.standardize_pf2``
+    # permutes B's *rows* so that its diagonal is maximal, pairing eigen-state i
+    # with component i, and permutes the columns of each projection to match.
+    # Permuting only B's columns here would move each diagonal entry off the
+    # diagonal and destroy that pairing, so the eigen-state axis is relabeled by
+    # the same permutation. Since P_k B is left unchanged by this relabeling
+    # (up to the component permutation), the reconstruction is untouched.
+    X.uns["Pf2_B"] = B[np.ix_(order, order)]
+
     if "projections" in X.obsm:
+        X.obsm["projections"] = np.asarray(X.obsm["projections"])[:, order]
         X.obsm["weighted_projections"] = (
             X.obsm["projections"] @ X.uns["Pf2_B"]
         ).astype(np.float32, copy=False)
diff --git a/scrise/tests/test_component_ordering.py b/scrise/tests/test_component_ordering.py
index c9335158..104ff3a6 100644
--- a/scrise/tests/test_component_ordering.py
+++ b/scrise/tests/test_component_ordering.py
@@ -126,7 +126,71 @@ def test_order_components_by_energy_reorders_weights_and_B():
     ordered = order_components_by_energy(adata)
 
     np.testing.assert_allclose(ordered.uns["Pf2_weights"], weights[order])
-    np.testing.assert_allclose(ordered.uns["Pf2_B"], B[:, order])
+    # Both axes of B are permuted: the component axis by the energy ordering,
+    # and the eigen-state axis by the same permutation (see below).
+    np.testing.assert_allclose(ordered.uns["Pf2_B"], B[np.ix_(order, order)])
+    np.testing.assert_allclose(
+        np.array(ordered.obsm["projections"]), projections[:, order]
+    )
+
+
+def test_order_components_by_energy_preserves_diagonal_B():
+    """``parafac2.utils.standardize_pf2`` leaves B with a maximal diagonal;
+    reordering components must not scramble it. Permuting only B's columns
+    would move entry (i, i) to (i, order^-1(i)) and destroy the pairing."""
+    rng = np.random.default_rng(4)
+    n_cells, n_genes, n_conditions, rank = 30, 20, 5, 4
+
+    A = rng.normal(size=(n_conditions, rank))
+    C = rng.normal(size=(n_genes, rank))
+    # A diagonally dominant B, as standardize_pf2 produces.
+    B = np.eye(rank) + 0.1 * rng.normal(size=(rank, rank))
+    weights = np.array([1.0, 5.0, 2.0, 9.0])
+    projections, _ = np.linalg.qr(rng.normal(size=(n_cells, rank)))
+
+    assert np.array_equal(np.argmax(np.abs(B), axis=0), np.arange(rank))
+
+    adata = make_mock_adata_from_factors(
+        A.copy(), B.copy(), C.copy(), weights.copy(), projections
+    )
+    ordered = order_components_by_energy(adata)
+
+    B_new = np.array(ordered.uns["Pf2_B"])
+    assert np.array_equal(np.argmax(np.abs(B_new), axis=0), np.arange(rank))
+    # The non-negative diagonal convention survives too.
+    assert np.all(np.diag(B_new) > 0)
+
+
+def test_order_components_by_energy_relabels_projection_columns():
+    """Permuting B's eigen-state axis is only valid if the projections are
+    permuted the same way, so that P_k B is unchanged up to the component
+    permutation."""
+    rng = np.random.default_rng(5)
+    n_cells, n_genes, n_conditions, rank = 35, 22, 6, 4
+
+    A = rng.normal(size=(n_conditions, rank))
+    C = rng.normal(size=(n_genes, rank))
+    B = rng.normal(size=(rank, rank))
+    weights = np.array([3.0, 1.0, 7.0, 2.0])
+    projections, _ = np.linalg.qr(rng.normal(size=(n_cells, rank)))
+
+    energy = np.abs(weights) * np.linalg.norm(A, axis=0) * np.linalg.norm(C, axis=0)
+    order = np.argsort(energy)[::-1]
+
+    adata = make_mock_adata_from_factors(
+        A.copy(), B.copy(), C.copy(), weights.copy(), projections
+    )
+    ordered = order_components_by_energy(adata)
+
+    np.testing.assert_allclose(
+        np.array(ordered.obsm["projections"]), projections[:, order]
+    )
+    # P_k B is unchanged, up to the component permutation.
+    np.testing.assert_allclose(
+        np.array(ordered.obsm["weighted_projections"]),
+        (projections @ B)[:, order],
+        atol=1e-5,
+    )
 
 
 def test_match_components_across_ranks_identifies_new_component():
diff --git a/scrise/tests/test_invariants.py b/scrise/tests/test_invariants.py
index ffc910cb..54c4a6a6 100644
--- a/scrise/tests/test_invariants.py
+++ b/scrise/tests/test_invariants.py
@@ -134,6 +134,12 @@ def test_order_components_by_energy_is_a_permutation(
     np.testing.assert_allclose(
         np.array(ordered.obsm["weighted_projections"]), before_wp[:, order], atol=1e-5
     )
+    # Both axes of B, and the projection columns, are relabeled by the same
+    # permutation, which is what keeps B's maximal-diagonal form intact.
+    np.testing.assert_allclose(np.array(ordered.uns["Pf2_B"]), B[np.ix_(order, order)])
+    np.testing.assert_allclose(
+        np.array(ordered.obsm["projections"]), projections[:, order]
+    )
     np.testing.assert_allclose(new_A.shape, A.shape)