Among the family of odd-parity topological superconductors derived from Bi₂Se₃, Cu_x(PbSe)₅(Bi₂Se₃)₆ (CPSBS) has been elucidated to have gap nodes. Although the nodal gap structure has been established by specific-heat and thermal-conductivity measurements, there has been no direct observation of the superconducting gap of CPSBS using scanning tunneling spectroscopy (STS). Here we report the first STS experiments on CPSBS down to 0.35 K, which found that the vortices generated by out-of-plane magnetic fields have an elliptical shape, reflecting the anisotropic gap structure. The orientation of the gap minima is found to be aligned with the bulk direction when the surface lattice image shows twofold symmetry, but, surprisingly, it is rotated by 30∘ when twofold symmetry is absent. In addition, the superconducting gap spectra in zero magnetic field suggest that the gap nodes are most likely lifted. We argue that only an emergent symmetry at the surface, allowing for a linear superposition of gap functions with different symmetries in the bulk, can lead to the rotation of the gap nodes. The absence of inversion symmetry at the surface additionally lifts the nodes. This result establishes the subtle but crucial role of crystalline symmetry in topological superconductivity.