No single-colour rectangle on a 12-gon

The twelve vertices of a regular dodecagon are each coloured red or blue independently with equal probability. Call a rectangle monochromatic if its four corners are vertices of the dodecagon and all four have the same colour. What is the probability that no monochromatic rectangle exists?

$29/128$. Rectangles inscribed in the dodecagon are exactly pairs of diameters, so one appears iff two diametrically opposite pairs are both red-red or both blue-blue. Counting the $6$ opposite pairs: no single-colour pair gives $2^6$, one gives $6\cdot2\cdot2^5$, one red-red plus one blue-blue gives $6\cdot5\cdot2^4$, for $928$ colourings out of $2^{12}$, i.e. $\tfrac{29}{128}$.