  The arithmetic triangular billiards are classically chaotic but have
Poissonian energy level statistics, in ostensible violation of the BGS
conjecture. We show that the length spectra of their periodic orbits divides
into subspectra differing by the parity of the number of reflections from the
triangle sides; in the quantum treatment that parity defines the reflection
phase of the orbit contribution to the Gutzwiller formula for the energy level
density. We apply these results to all 85 arithmetic triangles and establish
the boundary conditions under which the quantum billiard is \textquotedblleft
genuinely arithmetic\textquotedblright, i. e., has Poissonian level statistics;
otherwise the billiard is "pseudo-arithmetic" and belongs to the GOE
universality class
