I start from the knowledge that the connected sum of two projective planes is homeomorphic to the Klein bottle.
For the first projective plane I want to use a normal projective plane, e.g. an euclidean or affine plane with the line at infinity included. Plane A, set of points and lines.
For the second projective plane I want to use the bundle of lines and planes through a point, which is isomorphic (or homeomorphic whatever) to a projective plane. In some books such a bundle is also called a projective plane in an abstract sense. So this is plane or bundle B, set of lines and planes.
Further I choose A and B so, that the bearer point of B is a point in A. Dually the bearer plane of A is a plane in B.
Is it now possible to speak of the connected sum of A and B and consider A and B together as a Klein bottle?
Although their elements partly are of a different nature, this could be overcome in an abstract way, by mapping all elements e.g. to coordinate vectors?
Also it is perhaps possible to restrict the situation to the lines of A and the lines of B so that they are of the same nature?
When not, what would be the mistake or problem in imagining this?