  The amplitude of two-photon transitions between hyperfine states in
hydrogenlike ions is derived based on relativistic Dirac equation and second
order perturbation theory. We study angular and linear polarization properties
of the photon pair emitted in the decay of $2s$ states, where spin-flip and
non-spin-flip transitions are highlighted. We pay particular attention to
hydrogenlike uranium, since it is an ideal candidate for investigating
relativistic and high-multipole effects, such as spin-flip transitions. Two
types of emission patterns are identified: i) non-spin-flip transitions are
found to be characterized by an angular distribution of the type
$W(\theta)\sim1+\cos^2\theta$ while the polarizations of the emitted photons
are parallel; ii) spin-flip transitions have somewhat smaller decay rates and
are found to be characterized by an angular distribution of the type
$W(\theta)\sim1-1/3\cos^2\theta$ while the polarizations of the emitted photons
are orthogonal, where $\theta$ is the angle between photons directions.
Deviations due to non-dipole and relativistic contributions are evaluated for
both types of transitions. This work is the first step toward exploring the
effect of nucleus over the the angular and polarization properties of the
photon pairs emitted by two-photon transitions.
