Surface states of topological insulators feature chiral spin-momentum locking. When such states are used as weak link between two superconductors, their spin texture gives rise to the anomalous Josephson effect, i.e., to a φ₀ shift in the current phase relation. In this work, we explore the anomalous Josephson effect in junctions where the weak link is a HgTe nanowire. We observe a giant anomalous φ₀-shift of the current-phase relation, which we attribute to the fact that HgTe surface states feature a single Fermi contour. Moreover, by varying the orientation of the in-plane magnetic field, we obtain information about the spin texture in momentum space. In particular, we found that the spin is not exactly perpendicular to the momentum, but shows a significant deviation of 19 degrees. Our results establish the anomalous Josephson effect as a sensitive tool to probe the spin texture of chiral 2D systems.