Weak topological insulators are conventionally characterized by weak $\left(Z_2\right)$ indices and are usually distinguished from strong topological insulators by the appearance of an even number of surface Dirac cones. We show that this description misses an essential signed structure: weak topological surfaces with the same number of Dirac cones can either carry a nonzero net chirality or be chirality compensated. This distinction is invisible to the weak indices. We derive a bulk-to-surface projection theory in which the chirality of each surface Dirac cone is determined by the surface-projected orientation of the corresponding bulk band-inversion Dirac sector. As a minimal demonstration, we construct two time-reversal-invariant lattice models with the same weak indices and the same surface Dirac-cone locations, but with opposite and identical cone chiralities, respectively. The resulting net surface chirality directly controls whether magnetic surface responses add or cancel. Our results provide a chirality-resolved refinement of weak topological surfaces beyond the conventional weak-index classification.

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• Dirac fermions
• Surface states
• Topological insulators
• Transport phenomena