I have had a closer look at the circuit diagram on the Honda DVD
The compressor relay is turned on and off by the ECM/PCM (Engine Control Module/Power-train Control Module), better known as the engine ECU. This is not surprising, as the engine revs pick up when the compressor comes on (if you force the compressor on using the wire in initial post of "diagnosing aircon problems", the engine revs do not pick up).
There is a signal from the aircon controller to the engine ECU, and it is easy to check that signal ....
With engine running, aircon on Auto, target temperature on Lo ....
#1. you should be able to measure battery voltage (14 with engine running) by using a volt-meter between battery positive and the right-hand pin in the socket in the picture below (ignore the paper clip in the socket)
#2. If you also measure current between the two pins (instead of using a paper clip), you should see a small current.
I actually did #2 on the OP's car, and there was a current, which puzzled me, explanation right at the end of this post.
I am absolutely confident that the controller in the OP's car is NOT faulty in any way, a check of #1 would confirm it.
The reason why the engine ECU is not turning the aircon relay on, is because the wire from the left-hand pin goes via the "relay control module". The "relay control module" is the "block" in the picture below.
The wire runs in a harness which plugs into the under-side of the "relay control module". The entire relay block (with relay control module) can easily be lifted up by removing the two bolts that hold it down. There is a good schematic on the Honda DVD, I will upload it later.
There is then a signal wire from the "relay control module" to the "gauge control module". The latter is in the dash, underneath the gauges (speedometer etc). That module can also be accessed, not as easily as the "relay control module", but much more easily than the aircon controller.
So to summarise ...
A. there is a wire from the aircon controller, direct to the right-hand pin in the picture of the connector for the pressure sensor, and this is a good place to check.
B. there is a wire from the left-hand pin to a plug on the underneath of the "relay control module"
C. there is a wire from another plug underneath the "relay control module", which goes to the "gauge control module".
D. there are two wires ***ociated with the aircon, from the "gauge control module" to the engine ECU
E. there is a wire from the engine ECU to the compressor relay
From what I have seen of the OP's car, IMO the fault is in the "relay control module" itself, or in the wire to the "gauge control module", or in the "gauge control module" itself. The reason I say this is because I did test #2 and there was a current, so IMO there must be continuity between the aircon controller all the way to the "relay control module". But I would recommend test #1 to be absolutely sure that the aircon controller is not faulty.
Maybe somebody know where i should find this part, new or used or replacement. I want go to Italy, but without air condition it will be nightmare 
