diff --git a/Lib/codecs.py b/Lib/codecs.py
index 9490602..f6c2448 100644
--- a/Lib/codecs.py
+++ b/Lib/codecs.py
@@ -374,6 +374,11 @@
         """
         pass
 
+    def seek(self, offset, whence=0):
+        self.stream.seek(offset, whence)
+        if whence == 0 and offset == 0:
+            self.reset()
+
     def __getattr__(self, name,
                     getattr=getattr):
 
@@ -606,8 +611,8 @@
 
             Resets the codec buffers used for keeping state.
         """
-        self.reset()
         self.stream.seek(offset, whence)
+        self.reset()
 
     def __next__(self):
 
@@ -700,8 +705,10 @@
         self.writer.reset()
 
     def seek(self, offset, whence=0):
-        self.reader.seek(offset, whence)
-        self.writer.seek(offset, whence)
+        self.stream.seek(offset, whence)
+        self.reader.reset()
+        if whence == 0 and offset == 0:
+            self.writer.reset()
 
     def __getattr__(self, name,
                     getattr=getattr):
diff --git a/Lib/encodings/utf_16.py b/Lib/encodings/utf_16.py
index 5500c06..809bc9a 100644
--- a/Lib/encodings/utf_16.py
+++ b/Lib/encodings/utf_16.py
@@ -103,17 +103,23 @@
 
 class StreamWriter(codecs.StreamWriter):
     def __init__(self, stream, errors='strict'):
-        self.bom_written = False
         codecs.StreamWriter.__init__(self, stream, errors)
+        self.encoder = None
+
+    def reset(self):
+        codecs.StreamWriter.reset(self)
+        self.encoder = None
 
     def encode(self, input, errors='strict'):
-        self.bom_written = True
-        result = codecs.utf_16_encode(input, errors)
-        if sys.byteorder == 'little':
-            self.encode = codecs.utf_16_le_encode
+        if self.encoder is None:
+            result = codecs.utf_16_encode(input, errors)
+            if sys.byteorder == 'little':
+                self.encoder = codecs.utf_16_le_encode
+            else:
+                self.encoder = codecs.utf_16_be_encode
+            return result
         else:
-            self.encode = codecs.utf_16_be_encode
-        return result
+            return self.encoder(input, errors)
 
 class StreamReader(codecs.StreamReader):
 
diff --git a/Lib/encodings/utf_32.py b/Lib/encodings/utf_32.py
index f0b7709..c052928 100644
--- a/Lib/encodings/utf_32.py
+++ b/Lib/encodings/utf_32.py
@@ -98,17 +98,23 @@
 
 class StreamWriter(codecs.StreamWriter):
     def __init__(self, stream, errors='strict'):
-        self.bom_written = False
+        self.encoder = None
         codecs.StreamWriter.__init__(self, stream, errors)
 
+    def reset(self):
+        codecs.StreamWriter.reset(self)
+        self.encoder = None
+
     def encode(self, input, errors='strict'):
-        self.bom_written = True
-        result = codecs.utf_32_encode(input, errors)
-        if sys.byteorder == 'little':
-            self.encode = codecs.utf_32_le_encode
+        if self.encoder is None:
+            result = codecs.utf_32_encode(input, errors)
+            if sys.byteorder == 'little':
+                self.encoder = codecs.utf_32_le_encode
+            else:
+                self.encoder = codecs.utf_32_be_encode
+            return result
         else:
-            self.encode = codecs.utf_32_be_encode
-        return result
+            return self.encoder(input, errors)
 
 class StreamReader(codecs.StreamReader):
 
diff --git a/Lib/test/test_codecs.py b/Lib/test/test_codecs.py
index 1316572..5d6b545 100644
--- a/Lib/test/test_codecs.py
+++ b/Lib/test/test_codecs.py
@@ -1602,8 +1602,8 @@
                  "utf-32-le",
                  "utf-32-be")
         for encoding in tests:
-            with codecs.open('foo', 'w+', encoding=encoding) as f:
-                # Check if the BOM is written only once
+            # Check if the BOM is written only once
+            with codecs.open(support.TESTFN, 'w+', encoding=encoding) as f:
                 f.write(data)
                 f.write(data)
                 f.seek(0)
@@ -1611,6 +1611,42 @@
                 f.seek(0)
                 self.assertEquals(f.read(), data * 2)
 
+            # Check that the BOM is written after a seek(0)
+            with codecs.open(support.TESTFN, 'w+', encoding=encoding) as f:
+                f.write(data[0])
+                self.assertNotEquals(f.tell(), 0)
+                f.seek(0)
+                f.write(data)
+                f.seek(0)
+                self.assertEquals(f.read(), data)
+
+            # (StreamWriter) Check that the BOM is written after a seek(0)
+            with codecs.open(support.TESTFN, 'w+', encoding=encoding) as f:
+                f.writer.write(data[0])
+                self.assertNotEquals(f.writer.tell(), 0)
+                f.writer.seek(0)
+                f.writer.write(data)
+                f.seek(0)
+                self.assertEquals(f.read(), data)
+
+            # Check that the BOM is not written after a seek() at a position
+            # different than the start
+            with codecs.open(support.TESTFN, 'w+', encoding=encoding) as f:
+                f.write(data)
+                f.seek(f.tell())
+                f.write(data)
+                f.seek(0)
+                self.assertEquals(f.read(), data * 2)
+
+            # (StreamWriter) Check that the BOM is not written after a seek()
+            # at a position different than the start
+            with codecs.open(support.TESTFN, 'w+', encoding=encoding) as f:
+                f.writer.write(data)
+                f.writer.seek(f.writer.tell())
+                f.writer.write(data)
+                f.seek(0)
+                self.assertEquals(f.read(), data * 2)
+
 
 def test_main():
     support.run_unittest(
diff --git a/Misc/NEWS b/Misc/NEWS
index 54ebf59..27d16aa 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -54,8 +54,9 @@
 Library
 -------
 
-- Issue #6268: Fix seek() method of codecs.open(), don't read the BOM twice
-  after seek(0)
+- Issue #6268: Fix seek() method of codecs.open(), don't read or write the BOM
+  twice after seek(0). Fix also reset() method of codecs, UTF-16, UTF-32 and
+  StreamWriter classes.
 
 - Issue #8782: Add a trailing newline in linecache.updatecache to the last line
   of files without one.