How Many Ways to Arrange Marbles in a 6x6 Grid Without Color Repetition?

[gonzo]

Okay, say you have a 6x6 grid. You have 36 marbles, in 6 colors, with 6 of each color. You want to arrange the marbles on the grid so that no row or column contains two marbles of the same color.

How many ways are there to do this?

 

[Galileo]

I guess there are many approaches to this problem. Some more tedious than other.
I think the following will work:
First consider a particular solution. (Draw the grid with numbers 1 to 6 for example).

To what extend is this solution unique?
Clearly, switching any 2 marbles of different colors will no longer give a solution.
However, switching columns or rows will give different solutions, as will any permutation of the colors on the balls.
Are these the only possibilities?

(I`m not claiming this is the best way to do the problem, but it will work).