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In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $H_L:L^2\left([b_L,0]\right)\to L^2\left([0,b_R]\right)$ and $H_R:L^2\left([0,b_R]\right)\to L^2\left([b_L,0]\right)$ . These operators arise when one studies the interior problem of tomography. The diagonalization of $H_R,H_L$ has been previously obtained, but only asymptotically when $b_L\ne -b_R$ . We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizes $H_R,H_L$ explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.