diff --git a/seaborn/categorical.py b/seaborn/categorical.py
index c32acd0ad1..ee8d3c222d 100644
--- a/seaborn/categorical.py
+++ b/seaborn/categorical.py
@@ -46,7 +46,9 @@ class _CategoricalPlotterNew(_RelationalPlotter):
     semantics = "x", "y", "hue", "units"
 
     wide_structure = {"x": "@columns", "y": "@values", "hue": "@columns"}
-    flat_structure = {"x": "@index", "y": "@values"}
+
+    # flat_structure = {"x": "@values", "y": "@values"}
+    flat_structure = {"y": "@values"}
 
     _legend_func = "scatter"
     _legend_attributes = ["color"]
@@ -69,6 +71,23 @@ def __init__(
         # round of refactoring that moves the logic deeper, but this will keep things
         # relatively sensible for now.
 
+        # For wide data, orient determines assignment to x/y differently from the
+        # wide_structure rules in _core. If we do decide to make orient part of the
+        # _core variable assignment, we'll want to figure out how to express that.
+        if self.input_format == "wide" and orient == "h":
+            self.plot_data = self.plot_data.rename(columns={"x": "y", "y": "x"})
+            orig_variables = set(self.variables)
+            orig_x = self.variables.pop("x", None)
+            orig_y = self.variables.pop("y", None)
+            orig_x_type = self.var_types.pop("x", None)
+            orig_y_type = self.var_types.pop("y", None)
+            if "x" in orig_variables:
+                self.variables["y"] = orig_x
+                self.var_types["y"] = orig_x_type
+            if "y" in orig_variables:
+                self.variables["x"] = orig_y
+                self.var_types["x"] = orig_y_type
+
         # The concept of an "orientation" is important to the original categorical
         # plots, but there's no provision for it in _core, so we need to do it here.
         # Note that it could be useful for the other functions in at least two ways
@@ -87,16 +106,6 @@ def __init__(
         if not self.has_xy_data:
             return
 
-        # For wide data, orient determines assignment to x/y differently from the
-        # wide_structure rules in _core. If we do decide to make orient part of the
-        # _core variable assignment, we'll want to figure out how to express that.
-        if self.input_format == "wide" and self.orient == "h":
-            self.plot_data = self.plot_data.rename(columns={"x": "y", "y": "x"})
-            orig_x, orig_y = self.variables["x"], self.variables["y"]
-            self.variables.update({"x": orig_y, "y": orig_x})
-            orig_x_type, orig_y_type = self.var_types["x"], self.var_types["y"]
-            self.var_types.update({"x": orig_y_type, "y": orig_x_type})
-
         # Categorical plots can be "univariate" in which case they get an anonymous
         # category label on the opposite axis. Note: this duplicates code in the core
         # scale_categorical function. We need to do it here because of the next line.
diff --git a/seaborn/tests/test_categorical.py b/seaborn/tests/test_categorical.py
index aa7317525c..f6c1b9f3c3 100644
--- a/seaborn/tests/test_categorical.py
+++ b/seaborn/tests/test_categorical.py
@@ -1723,22 +1723,14 @@ def test_flat(self, flat_series, orient):
         ax = self.func(data=flat_series, orient=orient)
         _draw_figure(ax.figure)
 
-        cat_idx = 0 if orient == "v" else 1
+        cat_idx = ["v", "h"].index(orient)
         val_idx = int(not cat_idx)
 
-        axis_objs = ax.xaxis, ax.yaxis
-        cat_axis = axis_objs[cat_idx]
-
-        for i, label in enumerate(cat_axis.get_majorticklabels()):
-
-            points = ax.collections[i]
-            point_pos = points.get_offsets().T
-            val_pos = point_pos[val_idx]
-            cat_pos = point_pos[cat_idx]
+        points = ax.collections[0]
+        pos = points.get_offsets().T
 
-            key = int(label.get_text())  # because fixture has integer index
-            assert_array_equal(val_pos, flat_series[key])
-            assert_array_equal(cat_pos, i)
+        assert_array_equal(pos[cat_idx].round(), np.zeros(len(flat_series)))
+        assert_array_equal(pos[val_idx], flat_series)
 
     @pytest.mark.parametrize(
         "variables,orient",