Running 240v from two separate 120v receptacles
Don't hokey-doke this thing when it's so easy to do it right.  

You already have the right wires in the wall

You already have a dedicated circuit for your table saw.  I presume you don't plan on running both saws at the same time.  So we'll simply convert the dedicated table saw line to 240V and we're done.  

Let's talk about the difference between a 240V circuit and a 120/240V circuit.   A 240V circuit has 2 hots and a safety ground.  A 120/240V circuit has 2 hots, neutral and a safety ground, and is able to power both 240V and 120V parts of the machine - such as a dryer's motor/timer or an oven's light. 

A table saw will need 240V power.  It has no use for neutral. 

Convert the 120V saw circuit to 240V

The cable is converted from hot-neutral to hot-hot simply by marking the white wire with black tape to redesignate it a hot.

Now, it must be punched down onto a circuit breaker.

We could do some very difficult rearranging of panel space. However there's an easier way.  There is a 20A 2-pole breaker in the left side center.  We'll replace that with a 20/20 tied quadplex. Possibly a Siemens (ITE) Q22020CT2, which provides two tied 2-pole 20A breakers.  


The inner breakers are tied and also are common-trip.  I don't know what that existing 20A breaker serves, but it might need common trip.  So we place that on the inner. 
The outer breakers are tied, but do not guarantee common trip (they do promise common maintenance shutoff).  They are suitable for this saw.  


So for this saw circuit, remove the hot from the old saw breaker, and its partner former neutral (the one now marked with black tape).  Both those go on the two (outer) hots on the new breaker. 

By the way, whoever runted all your hots and neutrals to barely enough length to reach the breakers they're on now.... don't invite them back.  Every hot and neutral should be long enough to reach every breaker space in the panel.