Extremely high magnetoresistance (XMR) in the lanthanum monopnictides LaX (X=Sb,Bi) has recently attracted interest in these compounds as candidate topological materials. However, their perfect electron-hole compensation provides an alternative explanation, so the possible role of topological surface states requires verification through direct observation. Our angle-resolved photoemission spectroscopy data reveal multiple Dirac-like surface states near the Fermi level in both materials. Intriguingly, we have observed circular dichroism in both surface and near-surface bulk bands. Thus the spin-orbit-coupling-induced orbital and spin angular momentum textures may provide a mechanism to forbid backscattering in zero field, suggesting that surface and near-surface bulk bands may contribute strongly to XMR in LaX. The extremely simple rocksalt structure of these materials and the ease with which high-quality crystals can be prepared suggest that they may be an ideal platform for further investigation of topological matter.