  A measurement of the decay time dependent CP-violating asymmetry in $B_s^0
\to \phi\phi$ decays is presented, along with measurements of the $T$-odd
triple-product asymmetries. In this decay channel, the CP-violating weak phase
arises from the interference between $B_s^0$-$\bar{B}_s^0$ mixing and the
loop-induced decay amplitude. Using a sample of proton-proton collision data
corresponding to an integrated luminosity of $3.0\, fb^{-1}$ collected with the
LHCb detector, a signal yield of approximately 4000 $B_s^0 \to \phi\phi$ decays
is obtained. The CP-violating phase is measured to be ${\phi_s
=-0.17\pm0.15\mathrm{\,(stat)}\pm0.03\mathrm{\,(syst)}}$ rad. The
triple-product asymmetries are measured to be
${A_U=-0.003\pm0.017\mathrm{\,(stat)}\pm0.006\mathrm{\,(syst)}}$ and
${A_V=-0.017\pm0.017\mathrm{\,(stat)}\pm0.006\mathrm{\,(syst)}}$. Results are
consistent with the hypothesis of CP conservation.
