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Electronic structure of the layered perovskite La1.5Sr0.5CoO4 with a checkerboard Co2+/Co3+ charge order is studied, using the local-spin-density approximation plus Hubbard U calculations including also the spin-orbit coupling and multiplet effect. Our results show that the Co2+ ion is in a high spin state (HS, t(2g)(5)e(g)(2)) and Co3+ low spin state (LS, t(2g)(6)). Due to a small Co2+ t(2g) crystal field splitting, the spin-orbit interaction produces an orbital moment of 0.26 mu(B) and accounts for the observed easy in-plane magnetism. Moreover, we find that the Co3+ intermediate spin state (IS, t(2g)(5)e(g)(1)) has a multiplet splitting of several tenths of eV and the lowest-lying one is still higher than the LS ground state by 120 meV, and that the Co3+ HS state (t(2g)(4)e(g)(2)) is more unstable by 310 meV. Either the IS or HS Co3+ ions would give rise to a wrong magnetic order and anisotropy.