A two-species Lotka-Volterra competition-diffusion model with spatially inhomogeneous reaction terms is investigated. The two species are assumed to be identical except for their interspecific competition coefficients. Viewing their common diffusion rate $\mu$ as a parameter, we describe the bifurcation diagram of the steady states, including  stability, in terms of two real functions of $\mu$. We also show that the bifurcation diagram can be rather complicated. Namely, given any two positive integers $l$ and $b$, the interspecific competition coefficients can be chosen such that there exist  at least $l$ bifurcating branches of positive stable  steady states which connect two  semi-trivial steady states of the same type (they vanish at the same component), and at least  $b$ other  bifurcating branches of positive  stable steady states that connect  semi-trivial steady states of different types.