Two wonders on the same square is already an overkill. Three? That would be beyond description - so, it is fortunate that the famous La Scala opera house is located not on the Cathedral Square but rather 350 meters away from it, Victor Emmanuel galleria connecting the two. The connection is very much by design as it symbolizes the unity between the religious/spiritual soul of the city and the cultural/secular one.
There are not so many renowned opera houses in the world. Some (like Sydney's "ship with billowing sails") are celebrated for their unique architectural design, others - for their opulence of the interior or even scale of operations (the Metropolitan Opera in New-York has for decades presented 27 operas each year!). La Scala is in the league of its own when it comes to the reputation and quality of its performers because the who-is-who of the opera world has forever regarded being invited to sing there as the ultimate honour and recognition of their professional excellence. It's not by chance that one of the most sought after souvenirs one can buy in the La Scala shop is...a poster featuring that of another undisputed opera star. I said "poster" because that's what they formally are but "painting" might be a better word. On a personal note, one such refined and deeply psychological portrait of young Maria Callas has for decades been my most prized art-related possession.
All the above might explain why off-season tours in the building remain one of the hottest tickets in Milan. Rather inexpensive, they offer the visitors an opportunity to soak in the elusive magic of the place - to lounge in a sumptuous armchair and feast their eyes on the heavy curtain adorned with jewels, the ornate ceiling, the shining chandelier, and even more shining loges heavily decorated with gold... After a while, one realizes that the Guardian Angel of Arts keeps his vigil over the unique "Staircase" even in the off-season.