Non-perturbative coupling of photons and excitons produces hybrid particles, exciton–polaritons, which have exhibited a variety of many-body phenomena in various microcavity systems. However, the vacuum Rabi splitting (VRS), which defines the strength of photon–exciton coupling, is usually a single constant for a given system. Here, we have developed a unique architecture in which excitons in an aligned single-chirality carbon nanotube film interact with cavity photons in polarization-dependent manners. The system reveals ultrastrong coupling (VRS up to 329 meV or a coupling-strength-to-transition-energy ratio of 13.3%) for polarization parallel to the nanotube axis, whereas VRS is absent for perpendicular polarization. Between these two extremes, VRS is continuously tunable through polarization rotation with exceptional points separating crossing and anticrossing. The points between exceptional points form equienergy arcs onto which the upper and lower polaritons coalesce. The demonstrated on-demand ultrastrong coupling provides ways to explore topological properties of polaritons and quantum technology applications.