Badly ordered cycles of circle maps

A cycle of a circle map of degree one is badly ordered if it cannot be divided into blocks of consecutive points, such that the blocks are permuted by the map like points of a cycle of a rational rotation. We find the smallest possible rotation intervals that a map with a badly ordered cycle of a given rotation number and period can have. Moreover, we show that if one of those intervals is contained in the interior of the rotation interval of a map then the map has a corresponding badly ordered cycle.