If I understand this correct, yes, you can. So you've created a report with a default grouping on it and shared the report with the user. Now, you want to create another grouping, but not the default to the report, you just want the user to be able to choose that grouping if they want? If so, create the grouping you want and then share that grouping with the user. (you also have to give them access to be able to change the groupings in the report). When they run the report, they can choose the grouping drop-down and they will see the grouping down at the bottom of the grouping list under "shared with me".
