Spiral twisting offers additional opportunities for controlling the loss, dispersion, and polarization state of light in optical fibers with noncircular guiding cores. Here, we report an effect that appears in continuously twisted photonic crystal fiber. Guided by the helical lattice of hollow channels, cladding light is forced to follow a spiral path. This diverts a fraction of the axial momentum flow into the azimuthal direction, leading to the formation of discrete orbital angular momentum states at wavelengths that scale linearly with the twist rate. Core-guided light phase-matches topologically to these leaky states, causing a series of dips in the transmitted spectrum. Twisted photonic crystal fiber has potential applications in, for example, band-rejection filters and dispersion control.