With a HDHP + HSA, can I deposit HSA funds into any HSA account?

Yes, this is allowed. The only downside to this (other than one more line to fill out on your tax return) is that if you make HSA contributions outside of payroll your money is first taxed by FICA and Medicare taxes. These are bypassed when you contribute through payroll, and adds up to a 7.65% tax.

I think what @littleadv is advising might actually be a good idea, but it is not called a rollover. It would go like this:

1. Contribute everything through payroll. Keep all contributions in cash. Thereby bypassing the FICA + Medicare taxes.
2. Once or twice (or however often you want) each year, do a Transfer of Assets from your company HSA to a different HSA. Make sure to indicate that you are not closing the source HSA. This is different than a rollover.
3. After the two weeks, the money is transferred, and you can invest how you see fit.