The spin probe technique was systematically applied to study a family of high permeability and high free volume glassy polymers. Rotation correlation times τ(c) or frequencies v = 1/τ(c) of 2,2,6,6-tetramethylpyperidine-1- oxyl (TEMPO) were measured in amorphous teflons random copolymers of tetrafluoroethylene and 2,2-bistrifluoromethyle-4,5-difluoro-1,3-dioxole, polyacetylenes and polynorbornenes. Polymers distinguished by unusually high, for the glassy state, permeability and free volume exhibit large rotational mobility of TEMPO. The correlation times correspond to fast rotation of the spin probe previously observed only in rubbery polymers. Correlations between the frequency v and gas permeability and diffusion coefficients were observed. However, in some polymers, the spin probe's rotation rate is also sensitive to side-chain local mobility, which does not affect translational diffusion coefficients of gas molecules in polymers.