  We investigate the re-entrant tetragonal phase in the iron-based
superconductor Ba0 .76K0.24Fe2As2 by DC magnetization and thermoelectrical
measurements. The reversible magnetization confirms by a thermodynamic method
that the spin alignment in the re-entrant C4 phase is out-of-plane, in
agreement with an itinerant double-Q magnetic order [Allred et al., Nat. Phys.
12, 493 (2016)]. The Nernst coefficient shows the typical unusually large
negative value in the stripe-type spin density wave (SDW) state owing to the
Fermi surface reconstruction associated with SDW and nematic order. At the
transition into the re-entrant C4 tetragonal phase it hardly changes, which
could indicate that instead of a complete vanishing of the associated charge
order, the spin reorientation could trigger a redistribution of the charges to
form a secondary charge order, e.g. in form of a chequerboard-like pattern that
no longer breaks the rotational C4 symmetry.
