I don't see anything that would have me rushing for the chainsaw...though pics are always iffy when it comes to definate statements. Gums hold deadwood (generally) for a very long time, still for peace of mind sure remove it.
Concentrate on the roots and soil, I agree with Eric the taper is not normal makes me think the tree has been buried during either initial civils for the subdivision or latter landscaping grade changes...new paths, roadways driveways etc... Check very closely around the base of the tree look for unusual bark textures, depressions....sound the stem, probe the soil for the root crown should be only 100-150mm deep just out approx 300mm from the stem...any deeper then grade changes have occured.
Maybe a root crown inspection would help but try to get as much info before you go firing up the compressor.
As I know you know topping Eucs is nuts and pointless unless you make money from lopping.
I personally wouldn't be cutting any live foliage from the tree unless I believed particular limbs required reduction due to the risk they posed property or persons, from the photos I don't see that, but perhaps from the ground there might be cause for some reduction.
Despite not having great form and again Eric's probably right past cutting most likely explaination, it is a very pretty tree, well worth working with the owner to find management options that serve the tree and their concerns.