Question: Why don't changes in sunrise and sunset times both reverse at the solstice?

Answer: They would if the earth followed a circular orbit perfectly centered on the sun. However, our clocks differ from solar time in a complex manner, due to variations in the length of the day that arise from the tilt of the earth's rotational axis combined with its slightly elliptical orbit around the sun. For more in-depth discussions, try "analemma" or "equation of time" in a web search.