Tensol 12 as previously mentioned is designed specifically for this job and is the one to use. Getting a nice clean transparent join is not easy to achieve especially if you don't do this sort of thing every day. I found myself in exactly this situation a couple of years ago when I wanted a perspex lid for my own turntable. After much thought and several attempts at cutting the perspex cleanly and accurately (jigs, routers, super sharp cutters etc.) it became obvious that I wasn't going to get a decent result without using a lot of time and Perspex!
I found a superb company in Bootle Liverpool who make Perspex fabrications all the time and do display cases for shops, museums and the like. They have also made many turntable lids for people over the years and to be honest the cost was less than my buying the perspex and the Tensol 12 but, more importantly they did an unbelievable job.
I'm sure there will be a similar outfit in your area but if you want to check out the folks I used here's the details:
http://epsliverpool.co.uk/
Really helpful friendly people.
Cheers,
Steve.