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We study the local geometry of n dimensional manifolds which are equipped with two integrable distributions, one of dimension $r$ and one of dimension $s$, where $r$ and $s$ are allowed to be unequal. We call them para-CR structures of type $(k,r,s)$, with $k=n-r-s\ge 0$ being the para-CR codimension. When $r=s$ they are the real analogues of CR structures. In the general case these structures are the natural geometric setting in which to discuss the geometry of systems of ODE's, as well as the geometry of systems of PDE's of finite...