Exercise from Guillemin and Pollack's book:
Assume $X \pitchfork Z$, both compact and oriented, and prove directly from the definition that $$I(X,Z)=(-1)^{(\dim X) (\dim Z)}I(Z,X).$$ (I assume $X,Z$ are (sub)manifolds of a manifold $Y\subset R^N$.)
In this case $I(X,Z)$ is the (signed) number of points in $X\cap Z$, where a point $y$ is included with a plus sign if the orientation of $X$ and $Z$ (in that order!) "add up" at $y$ to the orientation of $Y$; otherwise $y$ is counted with a minus sign.
I can see that this holds from the picture in the particular case when $X,Z$ are two "independent" loops on the torus, but I don't know how to generalize this picture to higher dimensions and write a rigorous proof
Suppose $V$ is an oriented real vector space of dimension $n$, with positive ordered basis $B=(v_1,\dots,v_n)$. Let moreover $U$ and $W$ be two subspaces of $V$ such that $V=U\oplus W$, and suppose they are oriented, with positive ordered bases $B_U=(u_1,\dots,u_r)$ and $B_W=(w_1,\dots,w_s)$. Of course, we have $n=r+s$.
We write $B_UB_W$ the concatenation of $B_U$ and $B_W$, that is, the sequence $(u_1,\dots,u_r,w_1,\dots,w_s)$; this is clearly an ordered basis of $V$. We say that the decomposition $U\oplus W$ of the oriented vector space $V$ as a direct sum of two oriented subspaces is positive or negative if $B_UB_W$ is a positive or a negative ordered basis of $V$, respectively, that is, if the change of basis matrix $C(B,B_UB_W)$ has positive or negative determinant, respectively.
Now $C(B_UB_W,B_WB_U)=(-1)^{rs}$ and $C(B,B_UB_W)=C(B_UB_W,B_WB_U)C(B,B_WB_U)$. This implies a once that the "sign" of the decomposition $U\oplus W$ of $V$ is $(-1)^{rs}$ times the "sign" of the decomposition $W\oplus U$.