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How arbitrary can I make my choice?

I do know that connections must transform correctly under coordinate transformations in order to keep my vector still a vector if I transport it, but is that it? Are there other restrictions on how I can choose my connection?

For example, we see pictures of parallel transport on the two sphere and because we can see the 2 sphere embedded in 3-D space, we can "intuit" what parallel transport would be like on that sphere. However, am I confined to that choice? Could I define an affine connection on a two sphere (given no metric, so I don't have to worry about compatibility issues) that would parallel transport vectors completely unintuitively to how I "would" do it from my 3-D perspective? Could I make the vectors twist and turn in weird fashion?

It seems to me that the affine connection is quite arbitrary; however, I have also seen equations that link it to how basis vectors "twist and turn" as we move throughout the manifold, so I am confused on really how arbitrary it is.

Perhaps I am too reliant on bases? @_@