Answer
Yes, this makes sense.
Work Step by Step
The middle polarizer is not at right angles to the first polarizer, so light will pass through this combination.
The middle filter effectively rotates the plane of polarization of the light that passes through the first polarizer.
Similarly, the middle polarizer is not at right angles to the final polarizer, so light will pass through this combination.
In summary, adding a middle polarizer can enable light to pass through a pair of crossed polarizers.
