My experiments show that Houdini's rotations use global Euler angles: first about world X, then about world Y, then world Z (assuming XYZ order).
An alternative is Tait-Bryan [en.wikipedia.org] local Euler angles, as in heading/elevation/bank for airplanes.
Is there any built-in way in Houdini to rotate an object using those local Euler angles? (with a bit of math, I'm sure a custom node could be created, but it'd be great if it already exists!)

you can change the rotation order as you need, so ZXY may be better representation for your case

tamte
ZXY may be better representation for your case

That's closer, but I don't think it's quite the same. For instance if you set ZXY mode with (45 degrees of bank), then neither of the other axes (X or Y) rotates around the model's local Y axis to changing its heading.

Well, it's probably not that important – I can convert to/from quaternions.

– Gary

euler angles are always hierarchical, and you can visualize them using Gimbal mode
from your link Tait-Bryan angles seem just line euler angles with different rotation order, not sure it solves the gimbal lock issue, does it?

quaternions are always an option