I have a masters in EE (concentration in computers -- this was before Maryland had a Computer Engineering program) and I'm working towards another masters in CS (part time after work). I was also a failed student in the applied math phd program. I was a TA in some CS classes.
I've taken tons of EE, CS, and Math courses.
When I was an undergraduate/graduate EE student at Maryland I used to think somewhat poorly of CS students (I thought all they learned was programing and that they were basically people who couldn't handle being an engineering major) but as I've taken computer science classes, I've come to appreciate and respect the major more.
Basically, CS can have higher level math that is just as deep and rigorous as the math used in engineering classes -- they are just in different branches of mathematics.
CS courses tend to have math concentrated in logic, discrete math, number theory, abstract algebra, and numerical analysis.
EE courses tend to have math concentrated in analysis (real and complex etc), transforms (laplace, fourier etc), statistics/probabiliy (signal estimation, stocastic processes etc).
I have found a lot of CS classes very interesting and challenging -- especially in the Algorithms and Theory of Computations (finite automata, context-free languages, turing machines, undecidability, computation complexity and NP-completeness).
BTW, a lot of math courses sound basic but the higher level courses can be challenging. The basic linear algrebra course (eigenvalue/eigenvector, linear vector spaces, etc) is pretty straight forward. But if you take a deeper one, I would place it above "basic math". Understanding the proof for the rational and Jordan Canonical form was somewhat difficult for me as was the theory of a single linear transformation, dual vector spaces and multilinear algrebra (quotient spaces, bilinear forms and duality, direct sums and tensor products etc)
As an EE major, I tend to place it at the top -- because you can do anything with it (I'm basically a programmer!).

But I would think it would be difficult for a CS major to get an engineering type job.