Why are polynomial filters for gradients in the opposite sign?
If you look at the formula above the table you linked to, you see that the output values are computed like

$$Y^{\prime}_j=\sum_{n=-4}^{4}D_ny_{j+n}$$

In Matlab this could simply be implemented by a vector product (for one output value $Y^{\prime}_j$, where the vector $y$ contains the input values around the current index $j$):

yd = D'*y;


On the other hand, convolution of these sequences is defined by

$$Y^{\prime}_j=\sum_{n=-4}^{4}D_ny_{j-n}$$

(note the '-' sign in the index of $y_j$!). So if you want to use the 'conv()' function in Matlab, you need to flip the coefficients in the vector $D$.