How do I compute the DC component given the other part of the Fourier spectrum? I often see examples in textbooks where someone constructs a 2D Fourier spectrum of an image by fixing two (symmetric) pixels to a given value, and computes the inverse DFT to get a nice waveform image. I want to know how these images are created. It seems to me that they pick one pixel that would give them the desired spacing and orientation, fill in the symmetric opposite pixel with the conjugate value, and then somehow fill in the DC component.

This last part is what's confusing me. I want to answer the more general question: if you're given the non-DC frequency components of the Fourier spectrum of an image, how do you compute the DC component? I would be happy with an algorithmic solution, as I'm working on signals in Python with the scipy/numpy suite.
Not at all. By definition, the components are orthogonal.