  We study spherical quadrilaterals whose angles are odd multiples of pi/2, and
the equivalent accessory parameter problem for the Heun equation. We obtain a
classification of these quadrilaterals up to isometry. For given angles, there
are finitely many one-dimensional continuous families which we enumerate. In
each family the conformal modulus is either bounded from above or bounded from
below, but not both, and the numbers of families of these two types are equal.
The results can be translated to classification of Heun's equations with real
parameters, whose exponent differences are odd multiples of 1/2, with unitary
monodromy.
