One of the groups to welcome the SWP's ‘open letter' was The Commune, a group that split from the AWL last year. On its website you will find, among other things, material from the Dutch and German Communist Left. At their meetings there is an openness of approach which is very different from what you're likely to come across in a typical leftist meeting.
However, their response to the SWP's latest idea shows what tradition they are still attached to. They say "We welcome the spirit of the Open Letter, and would be interested to participate in discussions concerning left unity in general, or a conference in particular." While they do have their criticisms these are thoroughly constructive. They think the way the Socialist Alliance came together is a good example to follow. They say it is similar to the NPA in France, a group the character of which "is all to play for."
An article on the European elections looks at the number of votes cast for left-wing parties and says "we get the surprisingly high tally of 340,000, which makes you think about what might have been achieved had the left got its act together over the last ten years". Basically they want the Left to get its act together and they see themselves as part of that process, with, incidentally, getting votes in elections as a target worth aiming at.
They often talk of "communism from below" and "emancipatory communism", but as long as they think that they are part of the ‘left' these words will have little real meaning.