smc

Each path must go through either the second or the fourth quadrant. Each path that goes through the second quadrant must pass through exactly one of the points $(-4,4)$, $(-3,3)$, and $(-2,2)$. There is $1$ path of the first kind, ${\left(\genfrac{}{}{0ex}{}{8}{1}\right)}^2=64$ paths of the second kind, and ${\left(\genfrac{}{}{0ex}{}{8}{2}\right)}^2=28^2=784$ paths of the third type. Each path that goes through the fourth quadrant must pass through exactly one of the points $(4,-4)$, $(3,-3)$, and $(2,-2)$. Again, there is $1$ path of the first kind, ${\left(\genfrac{}{}{0ex}{}{8}{1}\right)}^2=64$ paths of the second kind, and ${\left(\genfrac{}{}{0ex}{}{8}{2}\right)}^2=28^2=784$ paths of the third type. Hence the total number of paths is $2(1+64+784)=\boxed{1698}$.