Maybe... but not quickly, and not with current technology
Other answers have discussed the obvious approach, directly changing the rotation rate. Clearly this would require immense amounts of energy and cause catastrophic damage. What about a different approach?
Tip it over
As we know, the Earth's obliquity, or Axial Tilt is around 23.5° and causes the seasons. What if we, somehow, increased the Axial Tilt until the Earth turned "upside down"? From the perspective of a point above the Sun's North Pole, this would give the appearance that Earth's direction of rotation had reversed. As with any changes of this magnitude, we would want to proceed slowly, probably over the course of millennia. I don't have the necessary physics background to even begin estimating the energy requirements, but they should be at least an order of magnitude smaller than trying to directly reverse the direction of rotation.
At first, I thought the direction in which we tilt the axis would have a large effect (centuries of Summer without a sunset for most of the northern hemisphere - with an equally long, sunless Winter for the southern hemisphere - when the Axial Tilt reached 90°). However, this would only be true if the axis of rotation were to somehow remain pointed directly at the Sun. Instead, I think there would be a long period of slowly-intensifying extremes of Winter and Summer, with the Summer Sun reaching higher in the sky with each passing year, until we reached a peak where the Sun would briefly be directly above the North Pole before resuming is annual spiral back down below the horizon.
One possibility would be an enormous thruster (or a smaller set) mounted at each pole. The thruster units, or sets, would be pointed in opposite directions and mounted on turntables so the thrust would operate in a consistent direction with respect to the Sun's axis of rotation or some other suitable point of reference. Yes, there is a very good chance this will not work; it may simply slide the Earth's crust across the core, which could have devastating consequences if the core's axis diverges much from the crust's... but... I say it's worth a try!