In the USA
When one party holds the White House and the other party holds the majority of the two houses in the Congress, is the government 1/2 one party, 1/2 the other party, or is it almost like the ideology of the party holding the White House?
For the US nationally, I think history is the guide. The party holding both Houses of Congress has primary control over the domestic agenda, or at least policy is done on their terms. When you look at Eisenhower and Nixon/Ford, the Democrats were in charge of domestic policy. The opposite goes for when Clinton was faced with a Republican Congress. His 1996 State of the Union was in deference to a new political paradigm. For the most part, foreign policy has largely been left to the President (and some of those appointed to particular positions). That's not to say there haven't been instances where Congress has tried to reign in the President.
As far as state government goes, the Legislature tells you the true nature of the state (discounting partisan gerrymandering), not the Governor. The tendency of Massachusetts to elect Republican Governors doesn't tell you much about the state. The fact that it routinely sends a massive Democratic Supermajority to the legislature tells you much more. States that are safe for one party will generally have a legislature that reflects that fact. Gubernatorial elections can often be about personalities and perceived checks and balances (I say perceived because many states have supermajorities that can easily override a veto). It could be argued that any state could elect of a governor of either party in theory (including the recent past). You cannot argue that any state could elect a legislative majority of either party. It's not possible for Republicans to win control of the Massachusetts General Court, nor is it possible for Democrats to win control of the Utah State Legislature. If we had parliamentary governments across the country, I think many states would've easily exceeded pre-2015 Alberta for one-party control.
