MAX-2-XOR-SAT: Why does the special case work? I'm a new user so I cannot respond directly to this post here. 

I'm confused about the answers to the question, namely that MAX-2-XOR-SAT is in $P$ iff each clause is of the form $(x_i \oplus \neg x_j)$ (or is it $(x_i \oplus x_j)$? There were two different answers). Couldn't you make a simple substitution $\neg x_i \rightarrow y_i$ such that this is always the case, so that you can get the special case always? For example, you could convert, $(x_1 \oplus \neg x_2) \wedge (x_1 \oplus x_3)$, to $(x_1 \oplus y_2) \wedge (x_1 \oplus x_3)$. 

Forgive me if there is something obvious I am missing...I'm new to this field.

Thank you!



EDIT

To clear up confusion, I'm asking if a substitution is made for all clauses with a matching variable. Take for example, 

$$(x_1 \oplus x_2) \land (\neg x_1) \land (x_1 \oplus \neg x_3)$$

By substituting $\neg x_3 \rightarrow y_3$ we get:

$$(x_1 \oplus x_2) \land (\neg x_1) \land (x_1 \oplus y_3)$$

Where each 2-clause is of the form $(x_i \oplus x_j)$. It seems that in this case it still remains in $P$. Which raises a new question, how do single clauses affect the complexity of the solution?
No, you can't make a substitution like that.  The resulting formula isn't equivalent to the original formula.

Consider

$$(x_1 \oplus x_2) \land (x_1 \oplus \neg x_2).$$

This formula is not satisfiable.  However, if we apply your substitution, we end up with the formula

$$(x_1 \oplus x_2) \land (x_1 \oplus y_2).$$

The resulting formula now has 6 variables and is satisfiable.  For example, you can set $x_1=\text{True}$, $x_2=\text{False}$, $y_2=\text{False}$.

The problem is that your substitution doesn't ensure that $y_i = \neg x_i$, so it doesn't preserve satisfiability.  (If you wanted to preserve satisfiability, you could do that by making your substitution and then adding clauses of the form $(y_i \oplus x_i)$ ... but while that would preserve satisfiability, it wouldn't preserve the maximum number of clauses that can be simultaneously satisfied, so it still wouldn't help for solving MAX-2-XOR-SAT.)