The sphere $S^n$ has a CW-structure consisting of two cells in each dimension, say $S^n=e^0_1\cup e^0_2\cup\cdots \cup e^n_1\cup e^n_2$. But specifying a CW-structure is more subtle than that, am I right? Because for each cell one has to give the concrete attaching map $\varphi:S^{k-1}\to S^{k-1}$ that one has in mind (here the $S^{k-1}$ on the codomain is the $(k-1)$-skeleton).
And even if we agree a priori that these attaching maps are homeomorphisms, there are two different kinds of them: those such that $\varphi_*:H_{k-1}(S^{k-1})\to H_{k-1}(S^{k-1})$ is the identity map, and those such that $\varphi_*$ is minus the identity. Or, as some like to say, those that preserve orientation and those that reverse it.
In class my professor was computing the boundary map of the cellular chain complex (in which the $k-$th group is free abelian of rank 2 with generators $e^k_1,e^k_2$) that arises from this CW-structure on $S^n$. He started saying we had to make an orientation convention: "$e^k_1$ and $e^k_2$ are both oriented the same as $S^k\subseteq S^n$, the attaching map for $e^k_1$ preserves orientation, and then for $e^k_2$ to be oriented in a consistent way, its attaching map reverses orientation". Since the boundary map is defined in terms of degrees of maps between spheres, this allowed him to conclude that \begin{align*} \partial e^k_1&=e^{k-1}_1+e^{k-1}_2 \\ \partial e^k_2&=-e^{k-1}_1-e^{k-1}_2\end{align*}
So my question is, why would I care that $e^k_2$ is oriented in a consistent way (whatever that means)? Isn't it true that if I take, say, all attaching maps to be orientation preserving, then I get a different CW-structure on $S^n$ (same cells, different attachings), but just as legitimate as the one my professor choose? In that case, if I understand correctly, the boundary map would be $\partial e^k_1=\partial e^k_2=e^{k-1}_1+e^{k-1}_2$, is that a problem?