The spin structure of head-to-head domain walls in Ni80Fe20 structures is studied using high-resolution photoemission electron microscopy. The quantitative phase diagram is extracted from these measurements and found to exhibit two phase boundaries between vortex and transverse domain walls. The results are compared with available theoretical predictions and micromagnetic simulations and differences to the experiment are explained, taking into account thermal excitations. Temperature-dependent measurements show a thermally activated transformation of transverse to vortex domain walls in 7 nm thick and 730 nm wide structures at a transition temperature between 260 °C and 310 °C, which corresponds to a nucleation barrier height for a vortex wall between 6.7×10−21 J and 8.0×10−21 J.