  We study how nonsymmorphic symmetries that commute with lattice translations
are reflected in the quasiparticle interference (QPI) maps measured by scanning
tunneling microscopy (STM). QPI maps, which result from scattering of Bloch
states off impurities, record the interference of incoming and scattered waves
as a function of energy and tip's position. Although both the impurity and the
tip generically break spatial symmetries, we find that the QPI maps provide
universal information on these symmetries. The symmetries impose constraints on
the relation between various momentum components of the Bloch functions. These
relations result in selection rules on certain momentum transfers in QPI maps.
We find that universal information is encoded in the absence of QPI signal, or
in the relative intensity of its replications. We show examples for
one-dimensional chains and an effective model of the layered compound $\rm
ZrSiS$. We discuss the implications of our theory in the analysis of observed
QPI of the Weyl semimetal $\rm TaAs$. Our theory is particularly relevant for
materials in rod and layer space groups, or when a correlated order parameter,
such as antiferromagnetism, enlarges the unit cell.
