Thanks for the feedback. We definitely have tested the site (and homepage particularly) for any technical issues - if they are there, no one has found them yet.
I did neglect to mention initially that we rank highly in Bing for our main keyword, so it seems to be Google specific. Many interior pages rank competitively in Google, if it was indeed a technical issue, wouldn't they also be affected? It seems like Google has a problem with our homepage only. Also, in the time that the bouncing has been happening, I have completely redesigned the homepage - and it didn't affect anything - same SERPS results, bouncing in and out.