I want to combine the following two sentences by the word "or".

1) The cone is circular in the cases its base is a circle.
2) The cone is elliptic in the cases its base is an ellipse.

Can I combine the above two sentences by "or" as follow:

The cone is circular or elliptic in the cases its base is a circle or ellipse.

I'd say something like the following:
The cone is circular or elliptic when its base is a circle or an ellipse, respectively.
(I suspect GM's answer below is aimed at the original sentence (above my comment)...a result of both adding comments at the same time! I have used the word 'respectively' to align 'circle' with 'circular' and 'ellipse' with 'elliptical' - my sentence is still a bit clunky, though.)

No. The lists don't logically align. Your new version suggests that the cone could be circular when the base is elliptical. (But, I suspect you suspected that - hence your question.)