Time reversal is a key component in reverse-time migration (RTM) and source localization using wavefield extrapolation. The successful implementation of time reversal depends on the time symmetry (reversibility) of the wave equation in acoustic and elastic media. This symmetry in time, however, is no longer valid in attenuative media, and attenuation is often anisotropic. Here, we employ a viscoelastic anisotropic wave equation that decouples the influence of energy dissipation and velocity dispersion. That equation helps compensate for anisotropic attenuation and restore the time symmetry by changing the signs of the dissipation-dominated terms in time-reversed propagation, while keeping the dispersion-related terms unchanged. We test the -compensated time-reversal imaging algorithm on synthetic microseismic data from a 2D transversely isotropic medium with a vertical symmetry axis (VTI). After back-propagating multicomponent data acquired in a vertical borehole, we image microseismic sources using wavefield focusing. The source excitation times are estimated by picking the maximum amplitude of the squared shear strain component at the source locations. Accounting for attenuation anisotropy produces superior source images and more accurate excitation times compared to those obtained without attenuation compensation or with purely isotropic attenuation coefficients. The algorithm is also applied to a modified BP TI model to investigate the influence of such factors as survey geometry, errors in velocity and attenuation, noise, and limited aperture.