An assembly for a turbine engine includes a combustor bulkhead, a swirler and a mounting strap. The swirler includes a mounting lug and a tubular swirler body that extends along an axis. The mounting lug projects radially out from the swirler body. The mounting strap is attached to the combustor bulkhead. The mounting lug is axially restrained between the combustor bulkhead and the mounting strap.