  We address the problem of the foundation of generalized ensembles in
statistical physics. The approach is based on Boltzmann's concept of orthodes.
These are the statistical ensembles that satisfy the heat theorem, according to
which the heat exchanged divided by the temperature is an exact differential.
This approach can be seen as a mechanical approach alternative to the well
established information-theoretic one based on the maximization of generalized
information entropy. Our starting point are the Tsallis ensembles which have
been previously proved to be orthodes, and have been proved to interpolate
between canonical and microcanonical ensembles. Here we shall see that the
Tsallis ensembles belong to a wider class of orthodes that include the most
diverse types of ensembles. All such ensembles admit both a microcanonical-like
parametrization (via the energy), and a canonical-like one (via the parameter
$\beta$). For this reason we name them ``dual''. One central result used to
build the theory is a generalized equipartition theorem. The theory is
illustrated with a few examples and the equivalence of all the dual orthodes is
discussed.
