The target and double spin asymmetries of the exclusive pseudoscalar channel
$\vec e\vec p\to ep\pi^0$ were measured for the first time in the
deep-inelastic regime using a longitudinally polarized 5.9 GeV electron beam
and a longitudinally polarized proton target at Jefferson Lab with the CEBAF
Large Acceptance Spectrometer (CLAS). The data were collected over a large
kinematic phase space and divided into 110 four-dimensional bins of $Q^2$,
$x_B$, $-t$ and $\phi$. Large values of asymmetry moments clearly indicate a
substantial contribution to the polarized structure functions from transverse
virtual photon amplitudes. The interpretation of experimental data in terms of
generalized parton distributions (GPDs) provides the first insight on the
chiral-odd GPDs $\tilde{H}_T$ and $E_T$, and complement previous measurements
of unpolarized structure functions sensitive to the GPDs $H_T$ and $\bar E_T$.
These data provide necessary constraints for chiral-odd GPD parametrizations
and will strongly influence existing theoretical handbag models.