Can rigor mortis change the anatomical position in which a person died?
Rigor mortis does not cause movement. It causes rigor (Latin for stiffness.)

This is only a comparison, but think of it this way. Superglue doesn't shrink or change the shape of the connection of the two things glued together; it just holds them there with powerful 'stiffness'. So imagine that at death, part of our muscle cells exuded superglue. You would get stiff (the muscle to bone/fascia connections would not change) but you would not move. Immediately after death, some muscles relax; most obviously the eyelids open partially, the mouth, hands, and sphincters relax (you won't have to pry a gold coin from someone's dead hands like in the movies.) But aside from general relaxation, you very closely stay in the position you were in when you die. 

Physiologically, this is roughly how muscles work: muscle cells are filled with filaments, actin and myosin (I'm leaving out tropomyosin and  troponin for brevity.) Contraction occurs when the myosin can bind to the actin, break the bond and reform it on the next spot on actin, and repeat, like crawling along the actin. ATP and Calcium ions are necessary for this crawling movement. In death, ATP is quickly depleted (aerobically then anaerobically), whereas Calcium floods the cell, and the bonds within each muscle cell which were formed when death occurred become stable. The muscle does not contract any more; it just stiffens. But since stiffening requires contraction in life, it's called contraction in death.

As a parallel, touch your fist to your shoulder. Without resistance, there is some contraction there. Now keep it there while someone tries to move it; that requires much more forceful contraction, but the position of your arm doesn't change (remember, you're keeping your fist to your shoulder.) All this is thanks to the number and stability of those tiny bonds between actin and myosin. Those tiny bonds develop and stabilize in death. But they don't last forever, hence rigor reverses.

Rigor mortis has been studied extensively for an astonishing number of reasons including forensic studies and how it affects the meat we eat. 

I have seen a lot of death; one of my duties was to "pronounce" people dead. Sometimes when someone died during the night in a nursing home, a nurse would wait until morning to call - what difference did it make, anyway? - and I saw bodies in rigor. They were not all drawn up as in a fetal position. The nurses would usually straighten the patients out before rigor set in to make it more dignified and easier to transport the patient. If rigor caused movement, they wouldn't be straightened yet stiff  (rigor is something one needs to document, as is lividity.)

Rigor Mortis