To add a formal answer to the question, changing the position of the variables in the regressor matrix $$ X=(X_1,\ldots,X_p), $$ where $X_j=(x_{1j},\ldots,x_{nj})'$, amounts to postmultiplying $X$ with a $(p\times p)$ permutation matrix $P$ that has a single entry 1 in each column $j$ that indicates the new column position of that regressor $X_j$.
This matrix $P$ is invertible, where $P^{-1}$ is equal to the transpose of $P$, $P^{-1}=P'$.
Thus, the OLS coefficient of the regression of $y$ on the transformed regressors is \begin{eqnarray*} \hat\beta_t&=&((XP)'XP)^{-1}(XP)'y\\ &=&(P'X'XP)^{-1}P'X'y\\ &=&P^{-1}(X'X)^{-1}(P')^{-1}P'X'y\\ &=&P^{-1}(X'X)^{-1}X'y\\ &=&P'(X'X)^{-1}X'y\\ &=&P'\hat\beta\\ \end{eqnarray*} Here, $P'$ is a matrix that permutes the row elements of $\hat\beta$, and hence permutes the entries of the coeffient estimator accordingly.