\textit is to \emph as \textbf is to...?

The idea behind \emph is that it provides a high level way for giving emphasis to a part of the text. High level in the sense it is “independent” of the actual implementation.

The default behavior of \emph is to use italics when in an upright context and upright when in an italics context, but this can be modified on a document’s basis (or by a package implementing a particular style). In particular, your question has no real answer: \emph and \textit bear no “abstract” relationship; the relationship is only at the default implementation level.

This is different from stating some part of text is in italics; for instance, theorems are commonly typeset in italics and the styles use \itshape for this, not \em (the declarative form of \emph). Similarly they use \bfseries for the theorem tag (or \scshape or whatever).

You're free to define as many similar commands as you want. If your style requires a sort of “strong emphasis”, you can define \strong as you please, maybe using \bfseries in normal context and \extrabfseries (if your font supports it, the name is hypothetical) in a \strong context. Before doing this, think deeply whether your readers will be able to appreciate the difference between \emph{word} and \strong{word} (which I think they won't).

There may be some confusion as to how \emph and \em are defined. From the latex kernel (cf latex.ltx):

\DeclareTextFontCommand{\emph}{\em}
\DeclareRobustCommand\em
        {\@nomath\em \ifdim \fontdimen\@ne\font >\z@
                       \eminnershape \else \itshape \fi}%
\def\eminnershape{\upshape}%

Basically, \emph (via \em) checks if it's already in "emphasis mode". If it's not, \itshape is executed; and if it is, \eminnnershape -- which, by default, is set to \upshape -- is executed. (\eminnershape can be modified via a \renewcommand instruction. E.g., running \renewcommand\eminnershape{\scshape}} would select small-caps for "inner" emphasized material.)

So it's not the case that \emph is just "more semantic" than \textit. Indeed, \emph is more semantic -- it lets you choose the method of typographic emphasis (the defaults are \itshape for "outer" and \upshape for "inner" material) -- but it also provides a way of providing emphasis within emphasized material.

Finally, just in case you're curious how \DeclareTextFontCommand is defined in the LaTeX kernel, here goes:

\def \DeclareTextFontCommand #1#2{%
  \DeclareRobustCommand#1[1]{%
    \ifmmode
      \nfss@text{#2##1}%
    \else
      \hmode@bgroup
       \text@command{##1}%
       #2\check@icl ##1\check@icr
       \expandafter
      \egroup
    \fi
                       }%
}