Carpentry and Woodworking - Biscuits?

I pulled a kitchen table out of storage that I now would like to use
I'll be re-finishing it
It needs a little help

It's got two drop downs going the long way on either side (as does the grain)
In the up position both sides appear to have cupped, enough so stuff almost slides off
The middle isn't as obvious, but has cupped also
On one end it's cracked a bit in the middle (longways, with the grain)

Can I cut these pieces in half the long way, flip them, and re-join them to sort of flatten them out a bit?

If so, would the best way to join them be with biscuits?

If so, and I don't have a biscuit tool, can I use a table saw, or other kind of saw/tool that I may have?

Thank you for any tips, tricks, info, or solutions

I know the basic design you have, as we have one, too. Are the edges square, or is there a relieved design on each corner? If the table is basically a rectangle, then after you take the top off and flip it, you will probably see where the original 3 or 4 pieces were joined. These will be good ripping points. What probably happened is they joined 3 or 4 pieces and didn't reverse the growth rings when they assembled them. Once you have the top off and ripped, you can join them, reversing the middle piece to keep down the bowing. Same thing with the wings.
Now, biscuits are the best way to join this together, but you can also rip a spline groove in all the internal joints about 3/8" wide and get a reversing spline in the hardwood flooring department (or make your own), joining all the main top pieces and the wing pieces separately clamping them together with several pipe clamps. If you rip the spline groove all the way through, you will have to hide it, so either stop short when ripping or be prepared to install a "breadboard" on the ends. This is preferred, since it will stop all future warping of the table ends. Hope it wasn't too wordy.

Thanks Larry
The outside edges are square
How do I rip a 3/8ths spline groove?

On your table saw set the blade at 3/8" from the fence to the outside of the blade. Run the edge through the saw, flip it end for end and run it again. This should give you a centered spline groove. After you make these two passes, you may have to run it again down the middle to clean it out. Just move the fence about a blade width closer to the blade to do this. Of course a test cut on scrap wood will give the exact spacing from the fence because I don't know the total width of the table top.

That's do-able
Thanks Larry