> the parameters for the basis function are being solved simultaneously
> with the betas?

Yes, it's just a typical regression analysis in 3dDeconvolve. Nothing fancy: basis functions are simply used to formulate regressors (independent variables).

> What am I missing? As a simple example:
>
> Stim 1: _|_________|__________
> Stim 2: ______|_________|_____
> Voxel : __/\___/\___/\___/\____
>
> The solver could find that B1 = 1, B2 = 0, and claim that the IRF is:
> __/\___/\__
>
> Or that B1 = 1, B2 = 1, and the IRF is:
> _/\__
>
> Both yield a perfect fit, and both are valid tents. So what other constraints
> need to be satisfied? Or am I completely wrong about the IRF being solved
> simultaneously with the GLM, in which case, what actually happens?

Actually there are infinite solutions in such an indeterminate system. Two issues with your example:

(1) No noise in the signal is assumed; and
(2) It's an experiment with a terrible design in the sense the two stimulus types are perfectly correlated, leading to the so-called multicollinearity problem in which you can't tease apart the signal among the regressors. Typically the occurrences of the events in an FMRI experiment are arranged in such a way either the sequence between the types is random or the interval among them is random, or both, so that the regressors tend to be less correlated with each other.

HTH,
Gang