\left and \right not working on parentheses within the same line when internal fractions are changed

This is somewhat similar to this question.

The values in the fractions matter because the boxes of the characters have different sizes. The p has a descender below the baseline which the x doesn't, thus when you swap them, the box of the denominator get a little bigger and TeX uses a larger delimiter to make that fit.

You have a few possibilities to work around that (basically the same ones I listed in the linked question):

• You can use a fixed delimiter size (\bigg or \Big, for instance):

  $$\Bigl(\frac{x}{p}\Bigr)\Bigl(\frac{p}{x}\Bigr)$$
  $$\biggl(\frac{x}{p}\biggr)\biggl(\frac{p}{x}\biggr)$$
  

• You can \raise the p so that TeX won't try to use a larger box:

  $$\left(\frac{x}{\raise0.35ex\hbox{$p$}}\right)\left(\frac{p}{x}\right)$$
  

• or you can add an invisible p next to the x so that the delimiter used will be the larger one:

  $$\left(\frac{x}{p}\right)\left(\frac{p}{\vphantom{p}x}\right)$$
  

• Or you can change change TeX's \delimiterfactor (and \delimitershortfall, which I didn't show here) and let TeX adjust the delimiters accordingly:

  $$\delimiterfactor=790
  \left(\frac{x}{p}\right)\left(\frac{p}{x}\right)$$
  
  $$\delimiterfactor=970
  \left(\frac{x}{p}\right)\left(\frac{p}{x}\right)$$
  

Full code:

\documentclass{article}
\begin{document}
% \delimitershortfall=5pt % Default
% \delimiterfactor=901    % Default

$$\left(\frac{x}{p}\right)\left(\frac{p}{x}\right)$$

$$\Bigl(\frac{x}{p}\Bigr)\Bigl(\frac{p}{x}\Bigr)$$
$$\biggl(\frac{x}{p}\biggr)\biggl(\frac{p}{x}\biggr)$$

$$\left(\frac{x}{\raise0.35ex\hbox{$p$}}\right)\left(\frac{p}{x}\right)$$

$$\left(\frac{x}{p}\right)\left(\frac{p}{\vphantom{p}x}\right)$$

$$\delimiterfactor=790
\left(\frac{x}{p}\right)\left(\frac{p}{x}\right)$$

$$\delimiterfactor=970
\left(\frac{x}{p}\right)\left(\frac{p}{x}\right)$$
\end{document}

Output:

Use \genfrac for this:

\documentclass{article}
\usepackage{amsmath}

\newcommand{\genlegendre}[3]{\genfrac{(}{)}{}{#1}{#2}{#3}}
\newcommand{\legendre}[2]{\genlegendre{}{#1}{#2}}
\newcommand{\dlegendre}[2]{\genlegendre{0}{#1}{#2}}
\newcommand{\tlegendre}[2]{\genlegendre{1}{#1}{#2}}

\begin{document}

\[
\legendre{x}{p}\quad\legendre{p}{x}\quad
\legendre{x}{x}\quad\legendre{d}{b}
\]

\end{document}

The variants \dlegendre and \tlegendre are analogous to \dfrac and \tfrac.