Well, in simple terms you can't imagine a wave function. Usually if your dealing with a vector space, the concept of orthogonal is merely the fact that the dot products between two such vectors equal 0. Heres the killer, in quantum mechanics you would be wrong to think that you can even imagine a physical interpretation of the abstract vectors that represent the state of a wave function. Much of the mathematical abstraction that you get with quantum mechanics were made up by rules that have no physical sense except in very specific cases.
In your case you are wondering why an even and odd function are considered orthogonal or even what it means. When two functions are orthogonal it means that the integral of the two functions product over a period equals 0. It has nothing to do with how you'd imagine something to be perpendicular in the traditional sense. That is a mathematical definition.
The reason why all odd functions are orthogonal to all even functions that are periodic can be explained by simple Fourier analysis. I'm sure wiki would have an ok explanation.
