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We investigate a periodic quantum graph in form of a square lattice with ageneral self-adjoint coupling at the vertices. We analyze the spectrum, inparticular, its high-energy behaviour. Depending on the coupling type, bandsand gaps have different asymptotics. Bands may be flat even if the edges arecoupled, and non-flat band widths may behave as $\mathcal{O}(n^j),\,j=1,0,-1,-2,-3$, as the band index $n\to\infty$. The gaps may be ofasymptotically constant width or linearly growing with the latter case beinggeneric.

High-energy asymptotics of the spectrum of a periodic square-lattice quantum graph