I think CRVaughan is probably right, that you’ll want to pattern the cut, not the elements themselves. If circular patterning the features doesn’t work in SW, you can always make the cuts as intersecting bodies that you can almost always pattern, then doing a combine at the end the cut away where they intersect.
Also you could try the wrap feature since your part is cylindrical. You won’t have control over the taper, but it will be even. Wrap sometimes doesn’t like certain sketches, but when it works it’s great. It might take a bit of work to get a pattern that repeats right with even spacing, though, since you’ll have to map the revolved surface to a 2D sketch (remember, circumference=pi*D).
I’m not quite understanding your intention when you say “The shape of the grip is symmetric in the top view, but asymmetric in the side view.” Which view is supposed to be symmetric? Do you have a sketch that shows your intention?