The individual mandate and cap-and-trade may have originally been "Republican" ideas in some technical sense, but they were adopted under duress. They never truly represented things that Republicans supported. The same was true of the Bush tax hike, which even at the time conservatives viewed as the work of an apostate. So it's only natural that they haven't supported any of these things under the Obama adminstration. They never really did, after all, and this time around they felt that flat-out opposition was politically feasible. So that's what we got.See Yglesias here. I forget the policy area (was it guaranteed income? health care? Someone in comments will help me, I'll be) but I recall that there's a White House tape of Richard Nixon explicitly saying to Chuck Colson or Bob Haldeman or someone that his pretty liberal alternative to a Congressional initiative was just for show, and that if Congress actually agreed to it he'd have to veto his own plan, or something to that effect. But usually this sort of thing isn't venal; it's just the regular, healthy, ebb and flow of how politics and policy works. If you're afraid that the other party is going to pass something that hurts your party's constituency groups, you try to formulate an alternative that appears to -- and may actually -- reach the same goals but protects those groups.
For what it's worth, that's I think that's the best way to understand Democratic "support" for the Iraq war in 2002-2003. Yes, there were some Democrats who really wanted an invasion. But for the most part, Democrats were formulating an alternative that, had they actually been able to set the agenda, they would not have carried out. That is: a policy of inspections and threats eventually leading to an invasion was preferred by many Democrats to immediate invasion, but had a Democrat been in the White House Iraq would likely have been on the back burner if at all possible. Just as it was during Bill Clinton's presidency.
All of which goes back to something we've known at least since David Mayhew, which is that it's often tricky to tell the difference between the positions that politicians take and the policy they (would) make.