No. Mars rotates in the same direction as Earth. The rotational direction is defined as the direction in which the thumb would point if the fingers of the right hand were curled in the direction of Mars's rotation. That direction is the same for both Mars and Earth. Interestingly, the direction of revolution is also the same for both planets.
This common direction (in which the right-hand thumb would point) identifies North of the ecliptic plane (the plane defined by the Earth's orbit). Most planets (both rotation and revolution) of our solar system as well as the Sun (rotation) share this direction due to their common origin from a rotating mass of particles and gas.
So Earth and Mars both rotate in the same sense (counterclockwise if you look down on the solar system from North of the ecliptic). The Sun will always rise on Mars in the East, with East in the same sense as East on Earth.
After learning the ropes in physics at Wabash College, IN, Suniti Karunatillake enrolled in the Department of Physics as a doctoral candidate in Aug, 2001. However, the call of the planets, instilled in childhood by Carl Sagan's documentaries and Arthur C. Clarke's novels, was too strong to keep him anchored there. Suniti was apprenticed with Steve Squyres to become a planetary explorer. He mostly plays with data from the Mars Odyssey Gamma Ray Spectrometer and the Mars Exploration Rovers for his thesis project on Martian surface geochemistry, but often relies on the synergy of numerous remote sensing and surface missions to realize the story of Mars. He now works at Stonybrook.