NPN as toggle switch
You could trace out the pneumatic "air circuits" easily enough and give a better picture. BUT there is a good chance that it works as follows:

Cylinders may be double acting with air being admitted at end "A" and exhausted at "B" or admitted at "B" and exhausted at "A". This could easily be controlled with 4 transistors, with each transistor activating an inlet solenoid at one end and an exhaust solenoid at the other end simultaneously. 

Q1  Inlet 1A, exhaust 1B
 Q2  Inlet 1B, exhaust 1A
 Q3  Inlet 2A, exhaust 2B
 Q4  Inlet 2B, exhaust 2A  

Cycling: Q1 on - pause - Q1 off, Q2 on - pause, Q2 off .. repeat
 will shuttle cylinder 1 to &amp; fro.

Simiarly Q3, Q4 control cylinder 2.



Added:

4 NPNs seems rather reasonable - you have only 4 inputs with two NPNs being on at a time. as above. But, ...

Instead of transistors you could use 1 x ULN2803 IC or similar. Datasheet here - 8 darlington drivers in one IC. 

Use 4 channels just as for transistors

Nominate one side of each solenoid as below.
  Connect either side of each solendoid to V+.
  Connect pin 10 of ULN2803 to V+
 Connect pin 9 of ULN2803 to ground.
 Go!

Q1 Inlet 1A, exhaust 1B 
Q2 Inlet 1B, exhaust 1A 
Q3 Inlet 2A, exhaust 2B 
Q4 Inlet 2B, exhaust 2A

You can pair up 2 ULN2803 channels for extra drive.