Indentation of the signature in the letter class

The reason for the apparent "conditional indentation" stems from the source code for \closing:

\newcommand{\closing}[1]{\par\nobreak\vspace{\parskip}%
  \stopbreaks
  \noindent
  \ifx\@empty\fromaddress\else
  \hspace*{\longindentation}\fi
  \parbox{\indentedwidth}{\raggedright
       \ignorespaces #1\\[6\medskipamount]%
       \ifx\@empty\fromsig
           \fromname
       \else \fromsig \fi\strut}%
   \par}

The indent of \longindentation is only executed if \fromaddress is non-empty. And, \fromaddress is defined when using \address:

\newcommand*{\address}[1]{\def\fromaddress{#1}}

So, in order for the indentation to work properly (for whatever reason), you need something in \fromaddress. Using \def\fromaddress{\null} works just as well.

you can also put

\longindentation=0pt    

in the preamble (i.e. before \begin{document}).