  We investigate the optical properties of interlayer excitons in heterobilayer
transition metal dichalcogenides where moir\'e pattern is introduced by
heterostrain, in comparison with that introduced by twisting (and/or lattice
mismatch). Besides being a cause of the moir\'e texture, strain also
effectively introduces a constant gauge potential on either electron or hole,
which shifts the dispersion of kinetic energy with respect to the excitonic
crystal momenta in the moir\'e superlattices.This leads to distinct exciton
mini-band dispersions and light coupling properties from the twisting induced
moir\'e, even if the excitonic moir\'e superlattice potentials have the similar
real-space profile for the two cases. For strain that breaks the three-fold
rotational symmetry at the atomic scale,the exciton wave packets trapped at the
superlattice potential minima have elliptically polarized valley optical
selection rules, in contrast to the circularly polarized ones in the twisting
moir\'e. We investigate the evolution of the excitonic mini-bands and the
optical dipoles of the bright states inside the light cones with the decrease
of the moir\'e periodicity, upon which the excitonic wavefunctions evolve from
localized wave packets to the extended Bloch states. Furthermore, moir\'e
exciton properties under the interplay of twisting and heterostrain are also
discussed.
