  We introduce an odd double affine Hecke algebra (DaHa) generated by a
classical Weyl group W and two skew-polynomial subalgebras of anticommuting
generators. This algebra is shown to be Morita equivalent to another new DaHa
which are generated by W and two polynomial-Clifford subalgebras. There is yet
a third algebra containing a spin Weyl group algebra which is Morita
(super)equivalent to the above two algebras. We establish the PBW properties
and construct Verma-type representations via Dunkl operators for these
algebras.
