# Asymmetric (in sign) cross-price derivatives in consumer-theory problem

I'm puzzled why, in the following optimal-choice problem, good $$x$$ depends on price $$p_y$$ but the opposite is not true. Should not the sign of $$\frac{\partial x}{\partial p_y}$$ be the same of $$\frac{\partial y}{\partial p_x}$$?

Optimal-choice problem: given prices $$p_x$$ and $$p_y$$ and income $$m$$ find the optimal demand function for $$x$$ and for $$y$$: $$$$\max_{x,y} \{x\cdot(y+6)\} \qquad \text{s.t. } p_x x+p_y y \leq m \tag{1}$$$$ after imposing $$MRS_{xy}=p_x/p_y$$ and substituting back in the budget constraint I find: $$$$x = \frac{m + 6p_y}{2p_x} \qquad\qquad y = \frac{m}{2p_y}-3 \tag{2}$$$$ so that $$sign\left(\frac{\partial x}{\partial p_y}\right) = sign\left( \frac{6}{2p_x}\right)\neq sign\left(\frac{\partial y}{\partial p_x}\right) = 0$$.

What I would like to understand better:

1. How can this happen? I remember having read somewhere in MWG that the price-derivative matrix should be (even!) symmetric. Can you provide me with some explanation and/or some reference? Alternatively, is there any mistake in the calculations?
2. Provided the situation described above can happen [i.e. $$sign\left(\frac{\partial x}{\partial p_y}\right)\neq sign\left(\frac{\partial y}{\partial p_x}\right)$$], I would be very happy if you can come up with some "real world" examples of that.

$$S_{lk}(p,w) = \partial x_l(p,w) / \partial p_k + \partial x_l(p,w)/ \partial w * x_k(p,w)$$
First term: uncompensated change in demand of good l due to a change in $$p_k$$ (when multiplied by $$\partial p_k$$)
Second term: effect of wealth change on demand of good l (when multiplied by $$\partial p_k$$) [https://scholar.harvard.edu/files/basilico/files/mwg_flashcards.pdf]