Yes and no... you're not going to have paternal X chromosome characteristics because the gamete contains only an X or Y chromosome (male or female sperm) -- but your X chromosome will have genes from both your grandmother and grandfather. The consequence of crossing over is that when you receive a chromosome from your mother, you don't simply inherit an "A" that she received from your grandfather or an "a" that she received from your grandmother... you get some unique combination of the two, Aa. Thus, the chromosome you inherit from your mother will be different than the ones your mother carries.

In that sense, random assortment (2^23 X 2^23) doesn't accurately reflect the genetic variation that occurs.