  Beam-target double-spin asymmetries and target single-spin asymmetries were
measured for the exclusive $\pi^0$ electroproduction reaction $\gamma^* p \to p
\pi^0$, expanding an analysis of the $\gamma^* p \to n \pi^+$ reaction from the
same experiment. The results were obtained from scattering of 6 GeV
longitudinally polarized electrons off longitudinally polarized protons using
the CEBAF Large Acceptance Spectrometer at Jefferson Lab. The kinematic range
covered is $1.1<W<3$ GeV and $1<Q^2<6$ GeV$^2$. Results were obtained for about
5700 bins in $W$, $Q^2$, \cthcm, and $\phi^*$. The beam-target asymmetries were
found to generally be greater than zero, with relatively modest \phicmsp
dependence. The target asymmetries exhibit very strong \phicmsp dependence,
with a change in sign occurring between results at low $W$ and high $W$, in
contrast to $\pi^+$ electroproduction. Reasonable agreement is found with
phenomenological fits to previous data for $W<1.6$ GeV, but significant
differences are seen at higher $W$. When combined with cross section
measurements, as well as $\pi^+$ observables, the present results will provide
powerful constraints on nucleon resonance amplitudes at moderate and large
values of $Q^2$, for resonances with masses as high as 2.4 GeV.
