  Double-spin asymmetries in the cross section of electroproduction of $\rho^0$
and $\phi$ mesons on the proton and deuteron are measured at the HERMES
experiment. The photoabsorption asymmetry in exclusive $\rho^0$
electroproduction on the proton exhibits a positive tendency. This is
consistent with theoretical predictions that the exchange of an object with
unnatural parity contributes to exclusive $\rho^0$ electroproduction by
transverse photons. The photoabsorption asymmetry on the deuteron is found to
be consistent with zero. Double-spin asymmetries in $\rho^0$ and $\phi $ meson
electroproduction by quasi-real photons were also found to be consistent with
zero; the asymmetry in the case of the $\phi$ meson is compatible with a
theoretical prediction which involves $s\bar{s}$ knockout from the nucleon.
