I'm adding a few circuits to my existing main panel. replacing some fat breakers with the slim doubles. Well I'm down to just a few open spots on the ground/neutral bar. Doing some research, it appears that it is OK to double up the ground wires (not the neutrals!).

The question is, just put them in together and tighten the screw or twist them together first?

Nothing wrong with twisting then, nor is it necessary.  In a hot panel I would probably not.

And it would be appropriate to double up the neutrals if:

The inspector would only want to know if the lug was rated for X number of Y size wires.

So you have a disconnect in front of this panel? If not, the neutral and ground are bonded at this panel, thus ground=neutral.

I have always twisted them since that is the way I was taught.

Yes, it's the main panel so they are bonded. But apparently starting in 2005 or 2008 the code was changed to specify that the neutrals can't be doubled up since they are actually carrying current, where as the ground normally isn't.

I'll be opening the main breakers, but not pulling the meter, so just the two exposed lugs at the top of the panel will be hot. I'll try to twist the grounds if I have room to work. I only need to free up a few extra spots.

I had to tear open the drywall under the panel to run a wire under my house for a mini split install. I'm sick of removing the bolted in reloading bench near the panel and doing drywall work so I'm running some wires into the crawl space for later projects at the same time. I'm adding a circuit w/ a timer for outlets for Christmas lights, wire for a sub panel going in the upstairs/attic to support future bathroom remodel (tub heater, fan heater, possible bidet, possible heated floor), and for wine and beer fridges when we remodel the kitchen next year. I'm not going to connect most of the wires in the panel until I'm ready for those projects.

Building inspectors don't like it.

My house has several ground wires doubled in the bus bar.
I have plenty of spots empty. But sone electricians don't really care too much.

you can double grounds if the panel lable allows.

may be easier to buy another ground bar.

Can't you share a neutral as long as the breakers for the shared circuits are tied together? Or is that something different?

different,

he is asking if you can put two neutral wires in one hole. You are thinking of a multiwire branch circuit which uses two hots and one neutral to run two circuits through one XX/3 wire with either a two pole or tied breakers.

That is a multi-wire branch circuit.

The neutral only carries any imbalance between the two legs so it is (theoretically) never more than one breakers full current.

Problems occur with loads with a poor power factor and non-sinusoidal current flow that do not 'balance' correctly.

Computer power supplies (though now improved) had horrid power factors and overloaded plenty of neutrals in commercial (office) type settings.

The narrow pulses from switching style power supplies react with the inductance in the distribution wiring to cause overloading of shared neutral.

This was one of the things that drove the requirement for power factor control on PC power supplies.

You can only put two wires in the bar  of a panel that is listed for that installation.

Neutrals may NEVER have two wires unless it happens to be a stranded conductor and within the allowed size range for the termination.

You can use 'adapters' to split up a larger stranded wire into multiple holes.
You cannot simply separate the stranding into multiple holes.

Watch out.

Many 'residential grade' panels are not rated for more than a 100 amp breakers and some are less than that.