We observe twinning of two-dimensional (2D) rhombic colloidal crystals of hard Brownian rhombic platelets. By contrast to square particles, which have higher symmetry but can also form rhombic lattices at high densities, each rhombic particle has a distinguishable bidirectional pointing axis. This key feature, which is not readily seen in rhombic crystals of square colloids, facilitates observations of different types of twinning: contact, polysynthetic, and cyclic. Moreover, we find that the twinned crystals are slightly offset spatially along their shared mirror line. In addition, the average pointing axis of the particles in a single crystal is also offset on average by a small angle, either clockwise or counterclockwise, from the average pointing axis of the rhombic lattice yielding a form of nonlocal chiral symmetry breaking. Because mirror lines between contact twins introduce only a small reduction in the total number of accessible states, compared to a perfect single crystal, twinning and piecewise linear defects are commonly observed. Thus, twinning, which is usually associated with complex compositions in certain minerals, also emerges in a simpler 2D system of entropically driven, hard, achiral objects.