I am going to tell you how to use the same cable you use to program your Arduino with to program a Picaxe. I am sure this could also be used the other way around too. I wish I did this before I paid for the Picaxe FTDI adapter.
Why do we use a hex inverter to program a Picaxe with the Arduino cable?
Well the signals coming from the Arduino FTDI cable are inverted from that of the Picaxe FTDI cable. That means they are the opposite. If you tried to program a Picaxe with the Arduino cable it wouldn't work because when the Arduino cable sends a 0 the Picaxe really needs a 1 and when the Arduino sends a 1 the Picaxe really needs a 0. So we use a hex inverter to invert the output so we can program the Picaxe. This walkthrough shows that you can invert the signals by reprogramming the FTDI cable, but I did not want to do that every time I switched in between a Picaxe and Arduino.
Resources we will be using:
What we use in the hex inverter data sheet is the connection diagram on page 1. On the Picaxe manual we use the diagram of the download circuit on page 7 as a reference.
The hex inverter takes 5v. Wire 5v to pin 14 and 0v to pin 7. Notice the notch, that is the front of the chip and the circle on the chip indicates pin 1. I used a battery to power this circuit because I could not upload the program using the FTDI power. It may work for you, but I suggest just going ahead and using batteries to supply the power. I used 4 1.2v batteries. Pin 1 of the hex inverter is where the TX from your Arduino FTDI cable gets connected to. The TX is the orange wire on the Arduino FTDI cable. The inverted signal then comes out of pin 2. So pin 2 will needs to be connected to the audio jack. Refer to Picaxe's manual. The diagram of the above view shows that the Serial in of the Picaxe is connected to the bottom right pin of the audio jack (above view remember. Picaxe manual page 7). So wire pin 2 of the hex inverter to the bottom right pin of the audio jack.
Next we can see that Serial out from the Picaxe is wired to the top middle pin of the audio jack. So next wire the top middle pin of the audio jack to pin 3 of the hex inverter. Then wire pin 4 of the hex inverter to the FTDI RX. The FTDI RX is the yellow wire on the FTDI cable.
Finally connect all grounds in the circuit. The ground from your battery, the ground from the FTDI cable and the ground from the audio jack which is the bottom left pin.
Ok you are ready to program. Use a audio cable adapter to plug in a Picaxe board to your newly made adapter. Now plug in the FTDI cable and program. Make sure everything is powered on.