  We develop a systematic theory of microwave-induced oscillations in the
magnetoresistivity of a two-dimensional electron gas, focusing on the regime of
strongly overlapping Landau levels. At linear order in microwave power, two
novel mechanisms of the oscillations (``quadrupole'' and ``photovoltaic'') are
identified, in addition to those studied before (``displacement'' and
``inelastic''). The quadrupole and photovoltaic mechanisms are shown to be the
only ones that give rise to oscillations in the nondiagonal part of the
photoconductivity tensor. In the diagonal part, the inelastic contribution
dominates at moderate microwave power, while at elevated power the other
mechanisms become relevant. We demonstrate the crucial role of feedback
effects, which lead to a strong interplay of the four mechanisms in the
nonlinear photoresponse and yield, in particular, a nonmonotonic power
dependence of the photoconductivity, narrowing of the magnetoresonances, and a
nontrivial structure of the Hall photoresponse. At ultrahigh power, all effects
related to the Landau quantization decay due to a combination of the feedback
and multiphoton effects, restoring the classical Drude conductivity.
