Most parts of the country that are likely to get severe earthquakes are not as likely to get severe tornadoes, and vice versa.
If a building is designed to meet (for instance) the California earthquake code, will that make the building more likely to withstand a severe tornado?
Conversely, if a building is designed to withstand a tornado, is it more likely to withstand an earthquake?
If the answer is no, how easy would it be to design a building to resist both moderate earthquakes and moderate tornadoes?
I don’t know about the engineering specifically, but in Wellington NZ many of the buildings have been earthquake proofed though things like windows that are designed to pop out rather than smash inwardly, flexible skyscrapers and use things like quake breakers - basically like putting the building on small rails so the ground can shake and the building will remain more steady. None of that sounds like it would cope with the strength of wind blowing and circling around buildings - although Wellington is an area of frequent high winds and buildings are built accordingly.
The big difference is roof attachment. Where big winds are expected - tornado/hurricane etc you need to tie the roof down to heavy underground masses - usually part of the building footing.
The tie-down is usually steel rods attached to the rafters inside the outer wall cavity.
For earthquake you need to stabilise the walls and if possible prevent them buckling. Quite often that’s done by horizontal steel ties inside the outer wall cavity.
To get the best of all worlds you should probably build your structure as a cage made of I-Beams set onto a concrete raft and footings.
The engineering is not particularly difficult for either, but there are rather different considerations in getting a building to survive shaking and getting it to survive sucking. Some techniques, such as strengthening roof-to-wall connections, may help in both instances.