Taking things apart to see how they tick is called reduction, but (re)assembling the parts is emergence.

When you reduce something to its components, you lose information on how the components work together. Emergence 'finds' that information back.

Compare differentiation and integration, which lose and gain terms respectively.

In some cases, I can imagine differentiating and integrating certain functions actually would even be a direct demonstration of reduction and emergence.

Yeah that’s a nice addition. However, remember that reassembling is synthesis, not emergence. Emergence is what you /may/ get by reassembling, but must not. We are talking about systems, thus, in the end, you are correct. It’s just that the terms seem to be a bit muddled.