HSA 6 month rule

The way the mathematics works, the answer is yes, so long as you do not exceed the annual HSA contribution limit as calculated based on whatever months you have HDHP coverage and not Medicare.

Yes, the wording says that you cannot contribute to your HSA once you are on Medicare, but in practice, the calculations are done on an annual basis, so you can contribute whenever you want, so long as you don't exceed your annual HSA contribution limit as calculated by TurboTax.

NOTE: Medicare does the start of your coverage retroactive 6 months ONLY if it has been more than 6 months since your turned 65. If it has been only 3 months since you turned 65, then the retroactive start will be only 3 months back.

In other words, Medicare will start your coverage in the month you turn 65, but won't look back more than 6 months.

The Journal of Accountancy has a nice three-question Q&A on this issue, especially since the IRS doesn't make some of the finer details clear in Pub 969.