Modules, bimodules and complexes with a rigid structure

We describe more or less new results on modules without selfextensions, that is on "tilting" and "cotilting" modules involved in equivalences and dualities between classes of modules. As we shall see, these modules may be the underlying left or right modules of reasonably small bimodules with many symmetries but also some unexpected anti-symmetries, for instance with respect to injective envelopes. Finally, we show that also right bounded complexes of projective modules, with as few as possible maps up to homotopy between their shifts, that is "partial tilting" complexes in the sense of J. Rickard, seem to inherit from the "partial tilting" modules a combinatorial structure