Communism is neither left nor right.

The Left/Right divide is about property. Should it be collectivized, or individually owned and traded, ie Socialism vs Capitalism. Communism, therefore, must be left-wing.

It is just a construct of governance.

Yes and no. Communism is also economic in nature.

While it was originally promoted by leftists, it can be either left or right.

You cannot have Right-Wing Communists.

Some communists are conservative and/or fascist (ex. China).

The PRC is socially conservative, yes. Economically, it is Socialist, though certainly not yet Lower-Stage Communist. This does not make China “fascist” or right-wing. It is a socially reactionary, economicaly progressive state.