Structural studies on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>ReCl</mml:mi> <mml:mn>6</mml:mn> </mml:msub> </mml:mrow> </mml:math> <mml:math

K$_2$ReCl$_6$ belongs to the antifluorite family and exhibits a sequence of structural transitions above the onset of magnetic order at $T_N$ = 12 K. Because of its 5d3 electronic configuration in an octahedral coordination, the ground state is a pure spin state without orbital degeneracy within the LS coupling scheme, but it can become Jahn-Teller active in the strong spin-orbit coupling limit described by the $jj$ coupling [S. Streltsov and D. I. Khomskii, Phys. Rev. X 10, 031043 (2020)]. While the structural transitions in K$_2$ReCl$_6$ are understood in terms of octahedral rotation and tilting, the possible impact of a Jahn-Teller distortion remains an open issue. We report on comprehensive crystalstructure studies by means of powder neutron and single-crystal x-ray diffraction on K$_2$ReCl$_6$ and on K$_2$SnCl$_6$. The latter material is used as a reference, because it exhibits the same sequence of structural transitions as K$_2$ReCl$_6$, but possesses a filled 4d shell ruling out a Jahn-Teller distortion. While the ReCl$_6$ octahedron in K$_2$ReCl$_6$ presents sizable distortions at intermediate temperatures, there is no such distortion persisting to low temperatures excluding a sizable Jahn-Teller effect. Studies on polycrystalline samples of Rb$_2$ReCl$_6$ and Cs$_2$ReCl$_6$, in which the structural transitions are suppressed due to the larger alkaline ionic radius, also do not find any indications for a Jahn-Teller distortion.