  We analyze the doubly differential electron momentum distribution in
above-threshold ionization of atomic hydrogen by a linearly-polarized
mid-infrared laser pulse. We reproduce side rings in the momentum distribution
with forward-backward symmetry previously observed by Lemell \textit{et al.} in
Phys. Rev. A \textbf{87}, 013421(2013), whose origin, as far as we know, has
not been explained so far. By developing a Fourier theory of moir\'{e}
patterns, we demonstrate that such structures stems from the interplay between
intra- and intercycle interference patterns which work as two separate grids in
the two-dimensional momentum domain. We use a three dimensional (3D)
description based on the saddle-point approximation (SPA) to unravel the nature
of these structures. When the periods of the two grids (intra- and intercycle)
are similar, principal moir\'{e} patterns arise as concentric rings
symmetrically in the forward and backward directions at high electron kinetic
energy. Higher order moir\'{e} patterns are observed and characterized when the
period of one grid is multiple of the other. We find a scale law for the
position (in momentum space) of the center of the moir\'{e} rings in the
tunneling regime. We verify the SPA predictions by comparison with
time-dependent distorted wave strong-field approximation (SFA) calculations and
the solutions of the full 3D time-dependent Schr\"{o}dinger equation (TDSE).
