  We show that a consequence of isospin symmetry, recently discovered in mirror
conjugated one-nucleon decays, can be extended to mirror-conjugated
$\alpha$-particles decays, both virtual and real. For virtual $\alpha$-decays
of bound mirror pairs this symmetry manifests itself as a relation between the
Asymptotic Normalization Coefficients (ANCs) of $\alpha$-particle overlap
integrals. This relation is given by a simple analytical formula which involves
$\alpha$-particle separation energies and charges of residual nuclei. For
bound-unbound mirror pairs, the ANC of a bound nucleus is related to the
$\alpha$-width of the mirror unbound level. For unbound mirror pairs we get a
new analytical formula that relates the widths of mirror resonances. We test
the validity of these analytical formuli against the predictions of a two-body
potential and of a many-body microscopic cluster model for several mirror
states in $^7$Li-$^7$Be, $^{11}$B-$^{11}$C and $^{19}$F-$^{19}$Ne isotopes. We
show that these analytical formulae are valid in many cases but that some
deviations can be expected for isotopes with strongly deformed and easily
excited cores. In general, the results from microscopic model are not very
sensitive to model assumptions and can be used to predict unknown
astrophysically relevant cross sections using known information about mirror
systems.
