Ultra-wide bandgap semiconductors are ushering in the next generation of high power electronics. The correct crystal orientation can make or break successful epitaxy of such semiconductors. Here it is discovered that single-crystalline layers of $alpha$-(AlGa)$_2$O$_3$ alloys spanning bandgaps of 5.4 - 8.6 eV can be grown by molecular beam epitaxy. The key step is found to be the use of m-plane sapphire crystal. The phase transition of the epitaxial layers from the $alpha$- to the narrower bandgap $beta$-phase is catalyzed by the c-plane of the crystal. Because the c-plane is orthogonal to the growth front of the m-plane surface of the crystal, the narrower bandgap pathways are eliminated, revealing a route to much wider bandgap materials with structural purity. The resulting energy bandgaps of the epitaxial layers span a range beyond the reach of all other semiconductor families, heralding the successful epitaxial stabilization of the largest bandgap materials family to date.