Catalyzes the exchange of L-carnitine for gamma-butyrobetaine. (R)-carnitine(out) + 4-(trimethylamino)butanoate(in) = (R)-carnitine(in) + 4-(trimethylamino)butanoate(out) Amine and polyamine metabolism; carnitine metabolism. Homotrimer. Belongs to the BCCT transporter (TC 2.A.15) family. CaiT subfamily.