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We investigate the re-entrant tetragonal phase in the iron-basedsuperconductor Ba0 .76K0.24Fe2As2 by DC magnetization and thermoelectricalmeasurements. The reversible magnetization confirms by a thermodynamic methodthat the spin alignment in the re-entrant C4 phase is out-of-plane, inagreement with an itinerant double-Q magnetic order [Allred et al., Nat. Phys.12, 493 (2016)]. The Nernst coefficient shows the typical unusually largenegative value in the stripe-type spin density wave (SDW) state owing to theFermi surface reconstruction associated with SDW and nematic order. At thetransition into the re-entrant C4 tetragonal phase it hardly changes, whichcould indicate that instead of a complete vanishing of the associated chargeorder, the spin reorientation could trigger a redistribution of the charges toform a secondary charge order, e.g. in form of a chequerboard-like pattern thatno longer breaks the rotational C4 symmetry.

Possible coexistence of double-Q magnetic order and chequerboard charge order in the re-entrant tetragonal phase of Ba0.76K0.24Fe2As2