diff --git a/prometheus_client/metrics.py b/prometheus_client/metrics.py
index b34ca817..95770ebd 100644
--- a/prometheus_client/metrics.py
+++ b/prometheus_client/metrics.py
@@ -27,12 +27,12 @@ def _build_full_name(metric_type, name, namespace, subsystem, unit):
     if subsystem:
         full_name += subsystem + '_'
     full_name += name
+    if metric_type == 'counter' and full_name.endswith('_total'):
+        full_name = full_name[:-6]  # Munge to OpenMetrics.
     if unit and not full_name.endswith("_" + unit):
         full_name += "_" + unit
     if unit and metric_type in ('info', 'stateset'):
         raise ValueError('Metric name is of a type that cannot have a unit: ' + full_name)
-    if metric_type == 'counter' and full_name.endswith('_total'):
-        full_name = full_name[:-6]  # Munge to OpenMetrics.
     return full_name
 
 
diff --git a/tests/test_asgi.py b/tests/test_asgi.py
index b2d9a70f..e31a3b6e 100644
--- a/tests/test_asgi.py
+++ b/tests/test_asgi.py
@@ -80,11 +80,11 @@ def get_all_output(self):
                 break
         return outputs
 
-    def validate_metrics(self, metric_name, help_text, increments):
+    def validate_metrics(self, metric_name, help_text, increments, unit=''):
         """
         ASGI app serves the metrics from the provided registry.
         """
-        c = Counter(metric_name, help_text, registry=self.registry)
+        c = Counter(metric_name, help_text, unit=unit, registry=self.registry)
         for _ in range(increments):
             c.inc()
         # Create and run ASGI app
diff --git a/tests/test_core.py b/tests/test_core.py
index f0b856a5..b14e340b 100644
--- a/tests/test_core.py
+++ b/tests/test_core.py
@@ -527,6 +527,11 @@ def test_no_units_for_info_enum(self):
         self.assertRaises(ValueError, Info, 'foo', 'help', unit="x")
         self.assertRaises(ValueError, Enum, 'foo', 'help', unit="x")
 
+    def test_name_cleanup_before_unit_append(self):
+        self.assertEqual(self.counter._name, 'c')
+        self.c = Counter('c_total', 'help', unit="total", labelnames=['l'], registry=self.registry)
+        self.assertEqual(self.c._name, 'c_total')
+
 
 class TestMetricFamilies(unittest.TestCase):
     def setUp(self):
diff --git a/tests/test_exposition.py b/tests/test_exposition.py
index 47c200f3..21b3f59f 100644
--- a/tests/test_exposition.py
+++ b/tests/test_exposition.py
@@ -59,6 +59,17 @@ def test_counter(self):
 # HELP cc_created A counter
 # TYPE cc_created gauge
 cc_created 123.456
+""", generate_latest(self.registry))
+
+    def test_counter_name_unit_append(self):
+        c = Counter('requests', 'Request counter', unit="total", registry=self.registry)
+        c.inc()
+        self.assertEqual(b"""# HELP requests_total_total Request counter
+# TYPE requests_total_total counter
+requests_total_total 1.0
+# HELP requests_total_created Request counter
+# TYPE requests_total_created gauge
+requests_total_created 123.456
 """, generate_latest(self.registry))
 
     def test_counter_total(self):