The Republic of Kiribati (pronounced "Kirr-ih-bass") is, by population, a very small nation with only 117,000 people. It consists of 33 islands, of which a third are inhabited, stretching across 2,400 miles of central Pacific Ocean near the equator and 1,300 miles along the International Date Line.
Kiribati became famous on the internet a few months ago for its unimaginative place naming practices. Yet it has another claim to fame, too. This both large and small nation is spread across four hemispheres: the northern, southern, eastern, and western. The CIA World Factbook identifies it as the only nation so endowed.
-via TYWIWDBI | Image: Google Maps
It would appear that both the UK and France would qualify. I think that the US would qualify for a southern hemisphere territory as a result of Samoa and the Aleutians and Saipan are in the eastern hemisphere. A lot would depend upon what qualifies as being part of a country.
https://en.wikipedia.org/wiki/Overseas_France says 'Overseas regions have exactly the same status as France's mainland regions', but there are no overseas regions in both the S and W hemispheres.
OTOH, French Polynesia is in the south Pacific, but it's "an overseas collectivity of France and its sole overseas country", and does not have the status of France's mainland regions.
To my surprise, the US does not have land in the S+E hemispheres. Looks like we did until we renounced claims to Funafuti and three other Tuvaluan islands in 1983.
If one includes Antartica, there might be more.