I have a suspicion:
1. MC looks like the Emperor's dead wife.
2. Hyojo isn't the dead wife's son, he just thinks he is. He is probably the son of a concubine, therefore making him second in line for the throne, unlike Hwita and his twin. The plan probably was to make Hyojo emperor because its hard to decide between two twins who'se to rule, but Hyojo turnt out to be a uhh.. waves hands Like that. (It could even be that there's some confusion on which kid was born first etc. Lots of options for fun drama.)
3. Hwita and Yuta are literal identical twins, so Yuta wears the mask because he looks so creepily like his bro. Little brothers tend to hate that.
4. Hyojo might actually know the lady who MC looks like isn't his actual mother and might actually know he's a concubines son. This might explain the mommy issues too, since what if Emperor didn't take a new empress after the one MC looks like died? Bam. Resentment. Even if its not HER fault, but the emperor's. (It would also explain a lot about why Hyojo is like he is. You don't get mentally healthy kids easily, when the parents are bad.)
5. Please please writer oh please, have Hyojo take MC's little sister as a concubine to try and strong arm her, only to show the family what happens when you play stupid games. Please. (The family ofcourse agreed because they will get free food or whatever.)