  Collisions of twisted particles --- that is, non-plane-wave states of
photons, electrons, or any other particle, equipped with a non-zero orbital
angular momentum (OAM) with respect to its propagation direction --- offer
novel ways to probe particle structure and interactions. In the recent paper
\cite{Ivanov:2019vxe}, we argued that resonance production in twisted photon
collisions or twisted $e^+e^-$ annihilation gives access to parity- and
spin-sensitive observables in inclusive cross sections, even when the initial
particles are unpolarized. Here, we explore these features in detail, providing
a qualitative picture and illustrating it with numerical examples. We show how
one can detect parity-violating effects in collisions of unpolarized twisted
photons and how one can produce almost $100\%$ polarized vector mesons in
unpolarized twisted $e^+e^-$ annihilation. These examples highlight the
unprecedented level of control over polarization offered by twisted particles,
impossible in the usual plane wave collisions.
