:::That's not right, there's no sign difference. You have R X F. "Newton" is a scalar factor within F, and "meter" is a scalar factor within R. The scalar factors commute, even though the underlying vectors do not. Example:
(2, 0, 0) X (0, 3, 0) = (0, 0, 2*3) = (0, 0, 3*2)
Likewise,
(1 m,0,0) X (0,1N, 0) = (0,0,1 m*N) = (0,0,1 N*m)