Yes they do but not without their well planted deception and toooo many "BIG TENT" less than conservative Republicans. There are some exceptions but the majority of Republicans took up the mantle of liberalism in the name of being compassionate.
The term "Compassionate Conservative" makes me want to puke. When we are called "mean" by liberals we should consider it a compliment.
Remember that Margret Thatcher was called the "Iron Lady". Nothing compassionate about her. Either shape up or ship out was her motto.