Further information:
I grabbed this from the popup auth window before it could reload me out.
<script type=”text/javascript”>location.href = “https://twitter.com/oauth/authenticate?oauth_token=MYTOKEN”;</script>
<b>Warning</b>: Cannot modify header information – headers already sent by (output started at /var/www/html/domain.com/wp-content/plugins/twitconnect/twitconnect.php:712) in <b>/var/www/html/domain.com/wp-includes/pluggable.php</b> on line <b>897</b>
So it seems that I am being put into a reauth=1 loop. This is a single user version of WP, not multi-user.