diff --git a/sqlparse/filters/reindent.py b/sqlparse/filters/reindent.py
index ea41ac6b..0c501015 100644
--- a/sqlparse/filters/reindent.py
+++ b/sqlparse/filters/reindent.py
@@ -93,8 +93,14 @@ def _split_statements(self, tlist):
                 tidx -= 1
             # only break if it's not the first token
             if prev_:
-                tlist.insert_before(tidx, self.nl())
-                tidx += 1
+                # Check that there is actual non-whitespace content before
+                # this keyword.  If all preceding tokens are whitespace
+                # (e.g. inter-statement blank lines that process() already
+                # handles), skip -- otherwise we double-count the separator.
+                _, before = tlist.token_prev(tidx, skip_ws=True)
+                if before is not None:
+                    tlist.insert_before(tidx, self.nl())
+                    tidx += 1
             tidx, token = tlist.token_next_by(t=ttypes, idx=tidx)
 
     def _process(self, tlist):
diff --git a/tests/test_format.py b/tests/test_format.py
index 0cdbcf88..8ffb4522 100644
--- a/tests/test_format.py
+++ b/tests/test_format.py
@@ -649,6 +649,37 @@ def test_insert_values(self):
             '     , (3, 4)',
             '     , (5, 6)'])
 
+    def test_multi_statement_idempotent(self):
+        # Reformatting should not add accumulating blank lines
+        # between statements.
+        f = lambda sql: sqlparse.format(sql, reindent=True)
+        sql = 'select a from b; select c from d'
+        once = f(sql)
+        twice = f(once)
+        assert once == twice, f'not idempotent:\n  {once!r}\n  {twice!r}'
+
+        # Also with keyword_case
+        f2 = lambda sql: sqlparse.format(
+            sql, reindent=True, keyword_case='upper')
+        once = f2(sql)
+        twice = f2(once)
+        assert once == twice
+
+        # And with comma_first
+        f3 = lambda sql: sqlparse.format(
+            sql, reindent=True, comma_first=True)
+        once = f3(sql)
+        twice = f3(once)
+        assert once == twice
+
+    def test_union_idempotent(self):
+        # UNION within a single statement should also be idempotent
+        f = lambda sql: sqlparse.format(sql, reindent=True)
+        sql = 'select a from b union select c from d'
+        once = f(sql)
+        twice = f(once)
+        assert once == twice, f'not idempotent:\n  {once!r}\n  {twice!r}'
+
 
 class TestOutputFormat:
     def test_python(self):