I think that any party that was able to consolidate power like the Republicans after the Civil War would probably be deemed "Radical" because of the unprecedented nature of their political reality. The Republicans' desire to bring African- Americans, newly freed slaves, into the social and political process was radical in its assertion. African- Americans had become the first group to experience a Constitutional rejection/ negation and inclusion/ affirmation. The idea of a previously silenced group becoming a political voice was, in its very nature, radical. Women were to experience this later on and young people even after that. Additionally, I would say that the use of the federal government in such a strong and defiant manner was also radical. In a nation where federalism was a guiding principle in balancing state and federal power, the Radical Republicans did not seem afraid to using the strength of centralized power to enforce their agenda. There was little in way of Southern state negotiation, as the federal government mandated that the states follow the policy and path the Reconstruction agenda and had little in way of voice about it.
I would say they were quite radical in both categories. The political changes that occurred right after the Civil War, including the abolition of slavery, giving freed slaves citizenship and black males the right to vote were very radical acts, undertaken by most Republicans, radical or not. But Radical Republicans, led by Senator Thaddeus Stevens, also wanted to punish the South with the Conquered Province Theory, denying them citizenship and treated them like occupied military districts. While this did not get adopted, it was certainly radical.
Economically, Stevens also championed the "40 Acres and a Mule" idea of distributing former plantations to former slaves, first started as a convenience by General William T. Sherman, and giving each family a mule to work the land. This idea was also too radical for mainstream Republicans and the President who refused to adopt it. The Freedman's Bureau, also strongly supported by the radicals, did manage to function for six years after the Civil War bringing education to former slaves in Southern States where it had once been illegal to teach them anything. 200,000 blacks achieved basic literacy in just six years, a radical economic, but also social act.