The cage is called a sleeve I think.

I have had three radios in my car, an OEM radio, a Kenwood single DIN which used a sleeve and a Kenwood double DIN which also uses a sleeve.

The sleeves are actually the same size as the OEM radio outside dimensions and when inserted can be made to locate in clips used to hold the OEM radio. The sleeves I have had butt against the lip that surrounds the radio opening in the radio/AC bezel, so the sleeve has to be released from the radio mounting clips, this can be a bit tricky as the clips holding the sleeve may not be obvious as there are a number of clips that can be bent to locate the sleeve. The sleeves come with a number of clips that can be bent to clip to the radio opening pocket, maybe mark the ones you use.

The sleeve if there is one, has to come out so that the radio/AC bezel can come off. This bezel is held on with clips and two screws which are located in the top face of the bezel radio opening. The screws actually can be removed prior to removing the sleeve. Remove the screws and with a plastic trim tool carefully pry off the bezel, start about 1.5" up from the bottom on both sides which is where the two clips are.

The square white plastic control rods locate in square holes in the back of the round selector *****, make sure they are correctly located radially or the ***** will not make the correct selection of heating or cooling. Prior to removing the bezel note the position of the selector ***** and replace it in the same position, do not rotate the square rods. These rods have swivel joints on them and are liable to sag or get out of line if they are touched, make sure they are in the square holes in the back of the ***** before you reassemble it all. Someone put it all together with one **** not coupled to the rod, a lot of time wasted on that mistake.

I think this time I will do the mod on the board location as I am over half way from being able to take it out again.

The above is what I noticed when I was and still am playing around with these parts.