Well the best course of action is looking at owls themselves
Owls have large spaces in their 14 neck vertebrae (twice as ours) to allow for the blood vessels to move while they twist their head. They also have blood "pockets" on the lower portion of their beak, allowing for oxygenation even when their head are bending the max 270 degrees. Next is their feet: owls are zygodactyl meaning they have 2 toes on the front and 2 on the back, however, due to having extremely flexible ligaments, many owls can move one of their back toes 180 degrees.
Now why did I waste all of this time talking about owls? Because that also would apply to your species.
To have good rotation on your head, you'll need more vertebrae and need additional space to fit in blood vessels along with the trachea. For the overall toes and fingers, elastic and flexible ligaments that allow for movement. The elbows will face decent structural changes in both bone and muscle alike to be nearly symmetrical from front to back, and I'm sure the patella must be absent in these.
Now: here's why I think it's not a great idea to do this in all joints: it compromises stability and there's decent issues with certain joints.
Let's start with the hip movement at the spine level: is it possible? Yes, but it puts sone strain on the organs in the area, squeezes them, so why strain your guts like that unless it's extremely necessary?
Next: elbows and knees. As far as I know, no animal can bend these 180,it's just not worth it, running becomes a mess, and there might be higher risks of fracture, as stepping wrongly might cause your joint to bend the opposite way, making balance a tricky activity, since you'd rely more on your muscles than on your bones to stand.
So summing up, some things might be great to swivel/ move better, such as the neck, fingers, toes and ankles (better at climbing), maybe the legs at the hip joint. Other than that, as far as I know, you might bring them more harm than good. For additional reference, look up contortionists, they're a much better example of a flexible humanoid that's stable.