I don't think it's especially helpful to compare Eisenhower with Obama. The times are different and the problems we face are different.
Sometimes a general consensus changes and popular policy suddenly becomes unacceptable or vice versa.
Example: Almost no elected official in the first half of the 19th Century supported a woman's right to vote. No elected official in the second half of the 20th Century believed women should NOT be allowed to vote. The consensus shifted and those who did not fall in line were swept aside.
Comparing Lincoln and Reagan becomes rather difficult if you don't consider individuals in the context of their time.
Reagan opposed the equal rights amendment (rightwinger!) but supported universal suffrage (leftwinger!). Lincoln opposed universal suffrage (rightwinger!) but supported ending slavery (leftwinger!). Who was more Conservative? Who was more Liberal? Those questions become kind of meaningless if you ignore the context of their times.
Think about even just the past forty years.
Richard Nixon, a Republican president, imposed wage and price controls. At the time, many people believed it was good public policy to have the government dictating the price of cotton cloth and tooth brushes. You want to know who oversaw the program? Dick Cheney and Donald Rumsfeld! I'm not making this up. Dick Cheney, the archetypal ultra right-winger, worked to impose price controls on the American economy.

I challenge you to find a sitting congressional representative who would support price controls (other than symbolic ones on gasoline or executive pay).
Again, sometimes the consensus changes and politicians must change with it.
That's why I don't find the Left/Right paradigm particularly helpful. What the "Left" or "Right" position is keeps changing. You can judge someone in their time as a lefty or a rightwinger but you can't judge people across time.