Q:

What are integration scale and differentiation scale?

In scale-adapted Harris detector, the scale adapted second moment matrix is defined by:
$$\mu(x, \sigma_I, \sigma_D) = \sigma_D^2\ g(\sigma_I) *\left[ \begin{array}{cc}
L_x^2(x, \sigma_D) &L_xL_y(x, \sigma_D) \\
L_xL_y(x, \sigma_D) &L_y^2(x, \sigma_D)
\end{array} \right]$$,
while $\sigma_I$ is the integration scale, and $\sigma_D$ is the differentiation scale. What are they? Why use them?
UPDATE
According to @Dima's answer, I think $\mu(x,\sigma_I, \sigma_D)$ is actually something like this,
$$\mu(x,\sigma_I,\sigma_D) = \sigma_D^2\cdot g(\sigma_I)*\left(g(\sigma_D)*\begin{bmatrix}I_x^2(x) &I_xI_y(x) \\
I_xI_y(x) &I_y^2(x)\end{bmatrix}\right)$$, where $I_x,I_y$ are the 1st order derivatives. Is it right? If so, why bother convolving twice? And why multiply $\sigma_D^2$?

A:

The $\sigma_I$ determines the scale level at which the Harris corners are computed. Coarser scales (higher values of $\sigma_I$) correspond to larger corners.
The $\sigma_D$ is effectively the window size, over which the derivatives are summed to generate the entries of the matrix. If $\sigma_D$ is too small, then the detector will be seriously affected by noise.  If it is too large, then small corners may wash out.  In the original non-scale-adapted Harris detector you only have $\sigma_D$.
By the way, this is the first time I see the terms "integration scale" and "differentiation scale", and I am not sure I like them. The "differentiation scale" is actually the Gaussian envelope, over which the derivatives are integrated. :)

