Defining a symbol that scales when used as a subscript

Instead of creating your own symbol, LaTeX has a predefined \textexclamdown that one can use. So, technically, you can do:

\usepackage{amsmath}
\newcommand{\antishriek}{\text{\raisebox{\depth}{\textexclamdown}}}

amsmath's \text will ensure that the correct size is used, while \raisebox puts the inverted exclamation at the baseline (if you want it that way; similar to what \rotatebox[origin=c]{180} would do for !).

---

In general, if you wish to have a command vary in its presentation depending on the location where it's used in math mode, then you can consider using \mathchoice (or the wrapper \mathpalette). \mathchoice has the following format:

\mathchoice{<material for display style>}
           {<material for text style>}
           {<material for script style>}
           {<material for scriptscript style>}

So, in your case, you could supply the formatting (sizing and rotation or whatever) to suit your needs. Here's an example of what that means: Consider the hypothetical \somesymbol:

\newcommand{\somesymbol}{\mathchoice
  {a}% \displaystyle
  {b}% \textstyle
  {c}% \scriptstyle
  {d}% \scriptscriptstyle
}

The above "symbol" definition displays (left is in default \displaystyle, while the right is in forced \textstyle, for comparison):

\[
  \somesymbol^{\somesymbol^{\somesymbol^{\somesymbol}}}
  \quad
  \somesymbol_{\somesymbol_{\somesymbol_{\somesymbol}}}
  \qquad\textstyle
  \somesymbol^{\somesymbol^{\somesymbol^{\somesymbol}}}
  \quad
  \somesymbol_{\somesymbol_{\somesymbol_{\somesymbol}}}
\]

Specific to your symbol, you may then use (as an example):

\newcommand{\antishriek}{\mathchoice
  {\rotatebox[origin=c]{180}{!}}% \displaystyle
  {\rotatebox[origin=c]{180}{!}}% \textstyle
  {\rotatebox[origin=c]{180}{\scalebox{.7}{!}}}% \scriptstyle
  {\rotatebox[origin=c]{180}{\scalebox{.5}{!}}}% \scriptscriptstyle
}

which displays as

\[
  \somesymbol^{\somesymbol^{\somesymbol^{\somesymbol}}}
  \antishriek^{\antishriek^{\antishriek^{\antishriek}}}
  \quad
  \somesymbol_{\somesymbol_{\somesymbol_{\somesymbol}}}
  \antishriek_{\antishriek_{\antishriek_{\antishriek}}}
  \qquad\textstyle
  \somesymbol^{\somesymbol^{\somesymbol^{\somesymbol}}}
  \antishriek^{\antishriek^{\antishriek^{\antishriek}}}
  \quad
  \somesymbol_{\somesymbol_{\somesymbol_{\somesymbol}}}
  \antishriek_{\antishriek_{\antishriek_{\antishriek}}}
\]

Here they are together, to show the relative sizing:

\[
  {\somesymbol\antishriek}^{{\somesymbol\antishriek}^{{\somesymbol\antishriek}^{{\somesymbol\antishriek}}}}
  \quad
  {\somesymbol\antishriek}_{{\somesymbol\antishriek}_{{\somesymbol\antishriek}_{{\somesymbol\antishriek}}}}
  \qquad\textstyle
  {\somesymbol\antishriek}^{{\somesymbol\antishriek}^{{\somesymbol\antishriek}^{{\somesymbol\antishriek}}}}
  \quad
  {\somesymbol\antishriek}_{{\somesymbol\antishriek}_{{\somesymbol\antishriek}_{{\somesymbol\antishriek}}}}
\]

Relevant:

• What is \mathchoice?

• The mysteries of \mathpalette

Here is a scalerel solution in which ¡ is scaled to the vertical footprint of the ! in the appropriate math style.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath, mathrsfs,scalerel}%
\newcommand{\antishriek}{\scalerel*{$¡$}{!}}

\begin{document}

\[\mathscr{P}^{\antishriek}\mathscr{P}^{!}\quad 
  \Delta_{\mathscr{P}^{\antishriek}}\Delta_{\mathscr{P}^{!}}\quad  
  \antishriek !\]%
\end{document} 

Why make things more comples than they are?

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath, mathrsfs}%
\newcommand{\antishriek}{\text{¡}}

\begin{document}

\[\mathscr{P}^{\antishriek}\quad \Delta_{\mathscr{P}^{\antishriek}}\quad  \antishriek \]%

\end{document}