  We consider multi-component quantum mixtures (bosonic, fermionic, or mixed)
with strongly repulsive contact interactions in a one-dimensional harmonic
trap. In the limit of infinitely strong repulsion and zero temperature, using
the class-sum method, we study the symmetries of the spatial wave function of
the mixture. We find that the ground state of the system has the most symmetric
spatial wave function allowed by the type of mixture. This provides an example
of the generalized Lieb-Mattis theorem. Furthermore, we show that the symmetry
properties of the mixture are embedded in the large-momentum tails of the
momentum distribution, which we evaluate both at infinite repulsion by an exact
solution and at finite interactions using a numerical DMRG approach. This
implies that an experimental measurement of the Tan's contact would allow to
unambiguously determine the symmetry of any kind of multi-component mixture.
