As I've noted before, President Obama is essentially an Eisenhower Republican.
As further support for this observation, I've pulled a few excerpts from the 1956 Republican Party platform.
While jealously guarding the free institutions and preserving the principles upon which our Republic was founded and has flourished, the purpose of the Republican Party is to establish and maintain a peaceful world and build at home a dynamic prosperity in which every citizen fairly shares.'Nuf said.
We are proud of and shall continue our far-reaching and sound advances in matters of basic human needs—expansion of social security—broadened coverage in unemployment insurance —improved housing—and better health protection for all our people. We are determined that our government remain warmly responsive to the urgent social and economic problems of our people.
The protection of the right of workers to organize into unions and to bargain collectively is the firm and permanent policy of the Eisenhower Administration.
Update 6/14/2015 Yet more support for the view that the moderate Republicans of the past are the Democrats of today:
Update 2/14/2016: And David Cay Johnson agrees that "back in the day, (Obama) would have been called a liberal Republican",