Lattice-mismatched bilayer graphene self-organizes into a moiré network of one-dimensional domain walls that conduct electrons with low dissipation, attractive for low-power electronics. We show these channels are not always straight: as the lattice relaxes to minimize strain, the network can spontaneously curve into chiral morphology. Atomistic simulations map a phase diagram in which the strain-flexibility balance selects one of three stable domain wall morphologies-straight, mono-chiral, or dual-chiral. Electronic structure calculations show that this morphology controls where low-energy electrons accumulate: straight channels concentrate states at the domain-wall-connecting nodes, while chiral channels shift that weight onto the domain walls themselves. This network geometric switch lets the same moiré material support either localized electronic hot spots or directional conducting channels-two strategies for guiding electrons in low-power graphene devices.