  The magnetic and structural phase transitions occurring in K$_2$ReCl$_6$ were
studied by macroscopic and microscopic techniques. Structural phase transitions
associated with rotations of the ReCl$_6$ octahedra lower the symmetry from
cubic to monoclinic, form ferroelastic domains, and are visible in
susceptibility, specific heat and thermal expansion measurements. In the
antiferromagnetically ordered state slightly below $T_{\rm N}$=12\,K these
domains can be rearranged by a magnetic field inducing a relative elongation of
the polydomain crystal parallel to the field of 0.6\%. At zero field the
magnetic structure in K$_2$ReCl$_6$ does not exhibit a weak ferromagnetic
component, but at large magnetic field a distinct magnetic structure with a
finite weak ferromagnetic component is stabilized. High magnetic fields
rearrange the domains in the crystal to align the weak ferromagnetic moment
parallel to the field. The altered domain structure with the crystal elongation
is abruptly suppressed at lower temperature but persists upon heating to well
above $T_{\rm N}$. However, heating above the lowest structural phase
transition and successive cooling restore the initial shape.
