What is "to talk out of both sides of one's mouth"

Mouth \Mouth\ (mouth), n.; pl. Mouths (mou[th]z). [OE. mouth, mu[thorn], AS. m[=u][eth]; akin to D. mond, OS. m[=u][eth], G. mund, Icel. mu[eth]r, munnr, Sw. mun, Dan. mund, Goth. mun[thorn]s, and possibly L. mentum chin; or cf. D. muil mouth, muzzle, G. maul, OHG. m[=u]la, Icel. m[=u]li, and Skr. mukha mouth.]

1. The opening through which an animal receives food; the aperture between the jaws or between the lips; also, the cavity, containing the tongue and teeth, between the lips and the pharynx; the buccal cavity.

2. Hence: An opening affording entrance or exit; orifice; aperture; as:

   1. The opening of a vessel by which it is filled or emptied, charged or discharged; as, the mouth of a jar or pitcher; the mouth of the lacteal vessels, etc.

   2. The opening or entrance of any cavity, as a cave, pit, well, or den.

   3. The opening of a piece of ordnance, through which it is discharged.

   4. The opening through which the waters of a river or any stream are discharged.

   5. The entrance into a harbor.

3. (Saddlery) The crosspiece of a bridle bit, which enters the mouth of an animal.

4. A principal speaker; one who utters the common opinion; a mouthpiece.

   Every coffeehouse has some particular statesman belonging to it, who is the mouth of the street where he lives.
   --Addison.

5. Cry; voice. [Obs.]
   --Dryden.

6. Speech; language; testimony.

   That in the mouth of two or three witnesses every word may be established.
   --Matt. xviii. 16.

7. A wry face; a grimace; a mow. Counterfeit sad looks, Make mouths upon me when I turn my back. --Shak. Down at the mouth or Down in the mouth, chapfallen; of dejected countenance; depressed; discouraged. [Obs. or Colloq.] Mouth friend, one who professes friendship insincerely. --Shak. Mouth glass, a small mirror for inspecting the mouth or teeth. Mouth honor, honor given in words, but not felt. --Shak. Mouth organ. (Mus.)

   1. Pan's pipes. See Pandean.

   2. An harmonicon.

      Mouth pipe, an organ pipe with a lip or plate to cut the escaping air and make a sound.

      To stop the mouth, to silence or be silent; to put to shame; to confound.

      To put one's foot in one's mouth, to say something which causes one embarrassment.

      To run off at the mouth, to speak excessively.

      To talk out of both sides of one's mouth, to say things which are contradictory.

      The mouth of them that speak lies shall be stopped.
      --Ps. lxiii. 11.

      Whose mouths must be stopped.
      --Titus i. 11.