One may instead consider the "double affine braid group" $DB_n$, $\pi_1(C_n(E))$, which is the configuration space of $n$ points on an elliptic curve (or what matters topologically is that it's a $S^1\times S^1$). One finds generators $B_n$ corresponding to loops which are contained in a contractible ball, and new generators $X_1,\ldots, X_n, Y_1,\ldots, Y_n$ corresponding to taking the $n$$i$th point of the configuration and running it around the the inside our outside ring of the torus. One computes relations that the $X_i$ commute, the $Y_i$ commute, and $T_iX_iT_i=X_{i+1}$, $T_iY_{i+1}T_i=Y_i$, and (Y_2,X_1):=Y_2X_1Y_2^{-1}X_1^{-1}=T_1^{-2}$.
So there is a hierarchy of degenerations. The top and bottom of the hierarchy are essentially symmetric in the variables $X$ and $Y$, in the precise sense that there is a "Fourier transform" automorphism swapping the variables, in both cases. Note that at the top of the hierarchy, the Fourier transform is just the order four automorphism of the elliptic curve which is the matrix $((0,1),(-1,0))$ in $PSL_2(Z)$, the mapping class group of the torus, and which can be seen in various elementary ways. At the bottom of the hierarchy, the Fourier transform is just swapping $x_i$ and $y_i$ and is related to Fourier transform of differential operators on an abelian group. In the middle, there isn't really a Fourier transform, because the symmetry was broken by degenerating the $Y_i$, but leaving the $X_i$ unscathed.
I'm not an expert on any of these things, and I can't say I've worked through the topological construction for all the cases, but I think, to answer Stephen's question, we can simply choose any lattice in $\mathbb{C}$, of rank $2$ (say $\Lambda=\mathbb{Z}1 \oplus \mathbb{Z}\mathbf{i}), and consider $\pi_1(\mathfrak{h}_reg/\Lambda^r)$, where $r$ is the rank of $\mathfrak{h}$. Then one quotients by the relations that loops around singular points of this quotient have order 2. For instance, let me explain a nice thing that happens for BC_n type.
BC_n non-affine braid group means we configurations of 2n distinct points in $\mathbb{C}\backslash\{0}$, such that $x$ is in the configuration if and only if $-x$ is in the configuration. We don't allow zero because that we want pairs of matched points. It's not hard to see that $\pi_1$ of that configuration space is the braid group of type $BC_n$, because you can choose the repesentative of each pair lying in the upper half plane, and you get the upper half plane, except that the $r$ and $-r$ are identified for all real points, and zero is excluded. This is a punctured plane.
Now the prescription above should lead you to consider $2n$ distinct points on the elliptic curve (or rather $S^1\times S^1$; I only say elliptic curve because people sometimes mean $\mathbb{C}^\times$ by torus...). However, now you have more points to remove. Not only zero, but all half integer points would correspond to a place where $x$ and $-x$ collide. So you get the usual picture of the torus from a first course in topology, except with half-integer points removed. A very fun exercise is to work out that you can again choose the representative of each pair which lies in, say, the lower left corner of the torus, where we cut the torus in half along the diagonal from upper left to lower right. But now again you have to identify some edges, and you get .... $\mathbb{CP}^1$ with four punctures (corresponding to the four half integer points of your real torus.) Now you get to choose five parameters: one parameter for non-affine hyper planes, meaning the $T_i$, and then one for each of the new poles you've introduced. Imposing the Hecke relations on all those you get the double affine Hecke algebra of type $BC_n$ (sometimes called $C^\vee C_n$ for historical reasons) with those five parameters. Notice that this process leads you to a very different presentation of the DAHA, where you de-emphasize the lattices, and emphasize the loops around the singularities. I understand that S. Sahi gave a similar presentation for all DAHA's coming from root systems, but with probably deeper motivations than these drawings on surfaces. I'm not sure of the reference where Sahi did this.
At least in this example, one again sees the Fourier transform as just the obvious move on the real torus.
