As can be seen, I made several attempts before I found a solution.
Here is the "finished" cube, which proves the solution although not very expertly crafted.
In each half four edges need to be joined, where the white sticky paper shows through.
I noticed two ways to mark out the folds, below is another one. In both cases, at first I could only make a parallelpiped shape. Then I managed to make a half-ish right cube from the image below, like a broken eggshell. However the triangle on each of the four sides projected too far, and would not make a cube with another equal part. So I reverted to the first arrangement, and bingo. Originally, I had not thought to extend the four edges of the marked square, which gives the needed folds.
As @Bass points out, there are two solutions: one where the shapes are folded forwards along the crease lines; the other when the folds are made backwards. So each solution is the inside-out version of the other.