Step 1: The 'Reversed' Layout.
A look at the circuit will tell you this is not a run of the mill JT configuration. Most obvious will be the fact that there is only a single coil involved (the 220uH) - we are using a second transistor (Q2) and C1 to take over the timing requirements. This allows us to use a wider range of coil values, as well as operate over a larger voltage range.
Besides reversing the charge / discharge order, this circuit also reverses the location of the driver transistor and the coil, but wait, that's not all! The transistors all have reversed polarities, and even the output voltage is reversed!
No, it isn't an error! Diode D1, the LEDs and the charged battery all have their polarities reversed! That's because this Joule Thief is configured as a voltage inverter. This arrangement was chosen due to its advantages for this kind of application.
To improve efficiency, the traditional JT relies on a fairly constant battery supply (over a millisecond or so) to give it a boost when it is delivering power. With the limited output from a Solar Cell, we have to store all its power in C2 and feed it into the Lithium in one big pulse, meaning the capacitor will be "empty" for the few critical millisecond, cancelling the 'kick' the normal JT requires to work well.
Our 'Reversed' JT circuit will work as a regular JT - without the 3v Lithium load, an input of 1.2v will light up the LEDs quite nicely. Not strictly necessary, the LEDs are there so you can SEE the system working, and also to prevent the battery overcharging.