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Deformed nuclei possess enhanced moments violating time reversal invariance
($T$) and parity ($P$). Collective magnetic quadrupole moments (MQM) appear in
nuclei with a quadrupole deformation (which have ordinary $T$,$P$-conserving
collective electric quadrupole moments). Nuclei with an octupole deformation
have a collective electric octupole moment, electric dipole moment (EDM),
Schiff moment and MQM in the intrinsic frame which rotates with the nucleus. In
a state with definite angular momentum in the laboratory frame, these moments
are forbidden by $T$ and $P$ conservation, meaning their expectation values
vanish due to nuclear rotation. However, nuclei with an octupole deformation
have doublets of close opposite parity rotational states with the same spin,
which are mixed by $T$,$P$-violating nuclear forces. This mixing polarises the
orientation of the nuclear axis along the nuclear spin, and all moments
existing in the intrinsic frame appear in the laboratory frame (provided the
nuclear spin $I$ is sufficiently large to allow such a moment). Such a
mechanism produces enhanced $T$,$P$-violating nuclear moments. This enhancement
also takes place in nuclei with a soft octupole vibration mode. In this paper
we present updated estimates for the enhanced Schiff moment in isotopes of Eu,
Sm, Gd, Dy, Er, Fr, Rn, Ac, Ra, Th, Pa, U, Np and Pu in terms of the
CP-violating $\pi$-meson--nucleon interaction constants
$\bar{g}_{0},\bar{g}_{1}$ and $\bar{g}_{2}$, the QCD parameter $\bar{\theta}$
and the quark chromo-EDMs. The implications of the enhanced $T$,$P$-violating
moments to the search for axion dark matter in solid state experiments are also
discussed, with potential alternative candidate compounds in which we may
expect enhanced effects suggested.