An electronic device such as a portable electronic device may have display with a capacitive touch sensor. A cover may have a front portion and a rear portion that bend along a bend axis. The front portion may overlap the display when the cover is closed. Conductive structures such as a patterned conductive layer having elongated strips of conductive material separated by gaps or another predetermined pattern may be formed in the front portion of a cover or other portion of a cover that overlaps the display and capacitive touch sensor when the cover is closed. Control circuitry in the device can gather capacitance images with the touch sensor and can detect when the cover has transitioned from open to closed or from closed to open from whether the predetermined pattern becomes present or becomes absent in the capacitance images.