Why is $\textbf{D}$ the response to $\textbf{E}$? In the text Wooten, equation 2.69 shows $\textbf{D}$ being the response to $\textbf{E}$ with $\epsilon$ as the response function:
$$
\textbf{D}(\textbf{r},t) = \int d\textbf{r}^{\prime} \int dt^{\prime} \epsilon(t,t^{\prime},\textbf{r},\textbf{r}^{\prime}) \textbf{E}(\textbf{r}^{\prime},t^{\prime})
$$
But he also makes the point that (Table 2.1) D is the externally applied field, and E is the total response from the externally applied field and induced polarization.
It seems to me like D and E should be switched in that integral, since it makes the total field a response to the applied field.
Am I missing some detail? Can the integral be written in the other way, with, for example with $\textbf{D}$ in the integrand and $\epsilon^{-1}$ as the response function?
Edit: I thought D = E + P (dropping constants) was only the way we write it because the electric dipole represented by P is opposite to the induced field. But this appears to lead to the integral as shown above since we write P ~ E, and by defining $\epsilon$ we can write 
$$
\textbf{D} = \textbf{E} + \textbf{P} = \epsilon \textbf{E}
$$
. But also from above, it seems like P (induced polarization) ~ D to first-order. Higher order terms establish P in a material is proportional to the total field (E) in the material.
 A: The integral reads: the $D$-field is the retarded and delocalised field as due to the application of the $E$-field on a material which is defined via the $ϵ$ Kernel. It's a standard problem of response theory. The link between $D$ and $E$ is called the constitutive relation, and you need an explicit expression for ϵ to break the self-consistency of the Maxwell equations in materials. This could be done either by measurement and tabulations, or by a microscopic calculation using condensed matter theory. The second third of the 20-th century has been dedicated to this problem, when people tried to calculate $\epsilon$ from quantum mechanics, with more or less success.
The generic form of the constitutive relations can be guessed from symmetry requirements. Note a complete version would also relate the D-field to the B-field (the one without monopole, verifying $\nabla\cdot B =0$), since nothing forbids this in principle (but some symmetries do). Perhaps a reading of this paragraph of the wikipedia encyclopedia might be helpful. All the definitions I'm using follow this page.
To now answer directly your question: you might want to invert the $E-D$ relation, and in general you can, but it's not really helpful because what you usually measure is the $D$ and $H$ fields. In fact, what you really measure are voltage drops and currents in materials, and those are related to $D$ and $H$, not $B$ and $E$. The way to see this is to simply starts from the conservation of current 
$$\partial\rho+\nabla\cdot j=0$$
with $\partial\rho\equiv\partial\rho/\partial t$. You can apply the gauge-transform 
$$j=j^{\prime}+\nabla\times H-\partial D \;;\; \rho=\rho^{\prime}+\nabla\cdot D$$
you will not alter the conservation of the current. Note that the gauge-transformation above is nothing but the usual Maxwell's equations as defined on the Wikipedia page cited above, so when you measure charges, you measure the $D-H$-fields. More precisely, you measure their integrals along path, loop, and volume.
In modern experiments using SQUID, you can measure the flux $\int B\cdot dS$ as well, but that's an other story.
