It is difficult to give a comprehensive answer in a fly-by HN comment.

Very brief answer: It was complicated. It is complicated.

Slightly less brief answer: Define reconciliation. I doubt any kind of perfect, ideal reconciliation was ever reached. It is not like there were two sides who agreed to something: there were many political actors, with different opinions. It was a mess, and it still is a mess. Many people who had first-hand experience of the war or the terror or its consequences remained bitter. The distinction between the Red and the White social sphere continued for a long time (Anecdote: there were Red and White country-wide sports unions SVUL and TUL until 1993.)

Some people remain bitter. The kind of people who like to have arguments about history and historical figures continue to debate the matter and legacy of Mannerheim and which was worse, the White terror or the hypothetical Red terror if the Reds had won.

Another anecdote. A couple of years ago a student chapter at one university put a portrait of Mannerheim on the wall of their club room. One professor made a very angry public complaint. I don't know what happened afterwards, but there were lots opinions in newspapers and social media, for a moment.

If we define "reconciliation" as "why there was no further violence, at least not much", one reason was that some of the political leaders on White pushed through torpparilaki (a land reform). SDP could participate in the elections, and due to efforts of some centrist politicians on White side, could once enter a government. Economy was good, the Great Depression wasn't as bad as in many other places. When Winter War happened, many regular people on the Red side felt their lives were improving and were distrustful of Stalin's clumsy propaganda moves (it helped that many Reds who "escaped" to the USSR during 1920s-1930s were shot or imprisoned with other ethnic Karelians and Finns in Stalin's purges; the word eventually got back that the USSR was not a paradise). The Reds were divided: non-communist socialists like SDP leader Väinö Tanner enjoyed electoral support among the working class, but were also quite hated by the pro-Soviet left. Maybe the White-aligned political police simply was very efficient at suppressing the would-be revolutionary left (some true believers on the Red side attempted to continue underground pro-Soviet / revolutionary efforts, not very successfully.)

Other part of the answer is that there was a lot of ...not-exactly-reconciliation for several decades.

Remember, some on the Red side continued with not necessarily reconciliatory political efforts? Some on the White side were actively not pleased about this at all, and some objected to any leftists being allowed to participate in politics at all. Feeling the legacy of their Independence War Victory was being squandered. In the 1920s/30s, for several years, some white extremists burned leftists printing presses, kidnapped and roughhoused lots of perceived communists, killed some, and the state apparatus turned a blind eye. Eventually they proceeded to kidnap bourgeois politicians not deemed anti-communist enough, including the ex-president Ståhlberg. At some point the danger of a genuine coup / intra-white civil war was real. (Google for Mäntsälän kapina.) In the end, after the parties and politics to the left of SDP were effectively banned, the extreme right lost its steam.

After the WW2, many outright communists and other on the Red side of left who had spent 1920s-1930s out of politics or in prison were able to enter the politics (thanks Finland mostly losing the WW2 to the Soviets) in force. In some respects, they got some amounts of revenge, when the most bourgeois party, Kokoomus, could not enter the government for decades because of Soviet objections. Google for Kekkonen and Finlandization.