The role of the right hemisphere (RH) in core language processes is still a matter of intense debate. Most of the relevant evidence has come from studies of gray matter, with relatively little research on RH white matter (WM) connectivity. Using Diffusion Tensor Imaging-based tractography, the current work examined the role of the two hemispheres in language processing in 33 individuals with Primary Progressive Aphasia (PPA), aiming to better characterize the contribution of the RH to language processing in the context of left hemisphere (LH) damage. The findings confirm the impact of PPA on the integrity of the WM language tracts in the LH. Additionally, an examination of the relationship between tract integrity and language behaviors provides robust evidence of the involvement of the WM language tracts of both hemispheres in language processing in PPA. Importantly, this study provides novel evidence of a unique contribution of the RH to language processing (i.e. a contribution independent from that of the language-dominant LH). Finally, we provide evidence that the RH contribution is specific to language processing rather than being domain general. These findings allow us to better characterize the role of RH in language processing, particularly in the context of LH damage.