How do you say 'Twisted' Congress power balance?

I like the term gridlock.

I know of no set phrase for this situation (which is odd as it happens a lot), but I would used "mixed". As in "The Obama adminsitration may have a harder time moving their agenda this session because they face a mixed congress."

Does Japan have a two-party system?

The U.S. does have a two-party system, and a bicameral Congress, but there is no well-known term for describing the situation where different parties control the Senate and the House of Representatives. Possible ways to express this idea are: "opposing majorities" or "split congress."

In English-speaking countries with parliamentary legislatures, such as Britain, Canada, and Australia, they do have a term, "hung parliament," but this seems slightly different than you describe. In these countries there are many political parties, and "hung parliament" refers to the case where no one party has a majority of seats in the legislature.

The most commonly used term in American English (this situation isn't really possible in other English-speaking legislatures) is "split control of Congress" or "a split Congress". Google for those terms and you'll find lots of references.

When referring to the United States (and by extension, other countries with presidential forms of government), Wikipedia refers to this as a divided government.

In countries such as France, with a semi-presidential system, the situation can arise where there is a president from one party, but a different party is the largest one in the National Assembly, leading to the likelihood of the prime minister also being of that party. This is referred to as cohabitation.

The United Kingdom is a parliamentary democracy like Japan. However, its upper house, the House of Lords, is (currently) appointed, contains a significant number of independent members, and has a composition which is not directly related to the balance of parties in the lower house, the House of Commons. In addition, the House of Commons can overrule the House of Lords in certain circumstances. Hence I'm not aware of any term to describe the situation in the question which applies to the UK.