Using angle-resolved photoelectron spectroscopy (ARPES), we investigate the surface electronic structure of the magnetic van der Waals compounds MnBi$_4$Te$_7$ and MnBi$_6$Te$_{10}$, the $n=$~1 and 2 members of a modular (Bi$_2$Te$_3$)$_n$(MnBi$_2$Te$_4$) series, which have attracted recent interest as intrinsic magnetic topological insulators. Combining circular dichroic, spin-resolved and photon-energy-dependent ARPES measurements with calculations based on density functional theory, we unveil complex momentum-dependent orbital and spin textures in the surface electronic structure and disentangle topological from trivial surface bands. We find that the Dirac-cone dispersion of the topologial surface state is strongly perturbed by hybridization with valence-band states for Bi$_2$Te$_3$-terminated surfaces but remains preserved for MnBi$_2$Te$_4$-terminated surfaces. Our results firmly establish the topologically non-trivial nature of these magnetic van der Waals materials and indicate that the possibility of realizing a quantized anomalous Hall conductivity depends on surface termination.
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