We report epitaxial growth of a-plane (1120) AlInN layers nearly-lattice-matched to GaN. Unlike for c-plane oriented epilayers, a-plane Al1-xInxN cannot be simultaneously lattice-matched to GaN in both in-plane directions. We study the influence of temperature on indium incorporation and obtain nearly-lattice-matched Al0.81In0.19N at a growth temperature of 760 °C. We outline a procedure to check in-plane lattice mismatch using high-resolution x-ray diffraction, and evaluate the strain and critical thickness. Polarization-resolved optical transmission measurements of the Al0.81In0.19N epilayer reveal a difference in band gap of ∼140 meV between (electric field) E||c -axis and E⊥c conditions with room-temperature photoluminescence peaked at 3.38eV strongly polarized with E||c, in good agreement with strain-dependent band-structure calculations.