After performing some more tests, changing the TLS setting to handheld didn't actually speed up things as much as I originally thought. Rather, it changed how the page appears to load. In proxy mode, the request takes a very long time, and then the page renders fairly quickly. In handheld mode, the total time is about the same, but the page renders much slower. However, I can still verify that HTTP access is much faster. I've re-configured our testing webserver and both HTTPS and HTTP.

I've found some diagrams that made the different settings on the handheld much more clear to me.
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So, it appears that if your website is HTTPS, it's encrypted with 3 layers. Two with SSL and one with triple DES. Does this sound accurate?

Is anyone else experiencing low performance on internal HTTPS sites?