  Photoionization cross sections, photoelectron angular-asymmetry parameters
and photoionization time delays for $A$@C$_{60}$ endohedral atoms are studied
with account for both the individual and combined effects of dipole static
polarization (DSP) of C$_{60}$ by the outgoing photoelectron, interior static
polarization (ISP) of C$_{60}$ by the ion-remainder, $A^{+}$, and atomic-core
relaxation of the encapsulated atom upon its ionization. It is unraveled that
the DSP effect is weak; it changes the phase of confinement-resonant
oscillations in $\sigma_{n\ell}$, $\beta_{n\ell}$ and $\tau_{n\ell}$ without
generally noticeable changes in their magnitudes, unless $\sigma_{n\ell}$
concentrates a relatively large part of oscillator strength of the continuum
spectrum near threshold. This is counter-intuitive in view of a large dipole
static polarizability of C$_{60}$, $\alpha > 800$ $a.u.$. Furthermore, it is
demonstrated that the DSP effect results in the transmission of a part of
oscillator strength of the continuum spectrum of $A$@C$_{60}$ into its discrete
spectrum. It is shown that the DSP effect is counteracted by ISP. Possible
reasons behind the made findings are provided. Photoionization of Xe@C$_{60}$,
Ne@C$_{60}$, H@C$_{60}$ and some hypothetical C$^{-}_{60}$ fullerene anions is
chosen as a case study. For Xe@C$_{60}$, the role of atomic-core relaxation of
the ionized encapsuled Xe$^{+}$-ion-remainder on the $\rm 4d$ photoionization
of Xe@C$_{60}$ is revealed to be of utter significance, as in free Xe. The
random phase approximation with exchange (RPAE) and generalized RPAE were used
in the study. The C$_{60}$ cage is modelled by a spherical attractive potential
of a certain inner radius, thickness and depth. Its dipole static polarization
potential is approximated by the Bates dipole static potential.
