What would be the effect of modeling a binary predictor in an OLS model as [-1, 1] instead of [0, 1]? I am using an OLS model to predict a continuous variable using several continuous predictors and one binary categorical predictor. I know that usually binary variables are modeled as [0, 1], but I am curious what the effect of giving a binary categorical predictor values of [-1, 1] would be. I had accidentally done the [-1, 1] coding in the first round of analysis, and after correcting the coding to [0, 1] I saw that the change in coding made the regression result similar but less extreme. I realize I don't fully understand what change this had on the model, and I'd appreciate insight into why it is so important to code the categorical variable as [0, 1].
[Edit] To clarify what I mean by the result being 'less extreme': I am fitting this model separately for each individual in our dataset, and then I do a wilcoxon signed-rank test across the coefficients from the binary variable (the coefficients do not meet the normality assumption for a t-test). The test is significant for the coefficients resulting from the [-1, 1] coding but not for the [0, 1] coding. Naturally I would like the test to be significant but now I feel that I cannot accept this result because it is arbitrarily affected by the coding. Is testing the coefficients resulting from the [-1, 1] coding somehow more or less legitimate? I am not sure which test result to accept, or whether I have approached the problem incorrectly.

If the model with $X$ coded as $\{0,1\}$ has the following form:
$$
y_i = \alpha_0 + \alpha_1 X_i + \varepsilon
$$
and model with $X^*$ coded as $\{-1, 1\}$ has the following form:
$$
y_i = \beta_0 + \beta_1 X^*_i + \varepsilon
$$
we can write the following equivalences:
$$
\beta_0 = \alpha_0 + \frac{\alpha_1}{2} \\
\beta_1 = \frac{\alpha_1}{2}
$$
$\alpha_0$ is the predicted mean of the outcome when $X=0$, and $\alpha_1$ is the difference between the predicted mean of the when $X=1$ and the predicted mean when $X=0$, i.e., the difference between the two predicted means. $\beta_0$ is the midpoint between the two predicted means, and $\beta_1$ is the difference between the predicted mean when $X=1$ and the midpoint. When other predictors are in the model, these interpretations involve holding those other variables constant (i.e., for a pair of units that were the same on all other covariates but differed in $X$).
The p-value on the test for $\beta_1$ will be identical to that for $\alpha_1$, but this will not be true for $\beta_0$ and $\alpha_0$.
