Although several laser-plasma-based methods have been proposed for generating energetic electrons, positrons and γ-photons, manipulation of their microstructures is still challenging, and their angular momentum control has not yet been achieved. Here, we present and numerically demonstrate an all-optical scheme to generate bright GeV γ-photon and positron beams with controllable angular momentum by use of two counter-propagating circularly-polarized lasers in a near-critical-density plasma. The plasma acts as a 'switching medium', where the trapped electrons first obtain angular momentum from the drive laser pulse and then transfer it to the γ-photons via nonlinear Compton scattering. Further through the multiphoton Breit-Wheeler process, dense energetic positron beams are efficiently generated, whose angular momentum can be well controlled by laser-plasma interactions. This opens up a promising and feasible way to produce ultra-bright GeV γ-photons and positron beams with desirable angular momentum for a wide range of scientific research and applications.