Did you check out the page on the Arcade Theatre recently?
If you did, perhaps you saw this recently added view from Ethereal Reality's post on Noirish Los Angeles that showed the 1910 Pantages (now the Arcade) sticking up as if it were a highrise on what was then a fairly low skyline on the 500 block of Broadway.
If you looked down at the bottom of this detail (which you can click on to enlarge) you find yourself looking at the east side of the 600 block of Hill St.
And look at the arched entryway right above the parked car. This is perhaps the only good view in existence of the Shamrock Theatre, 608 S. Hill. It opened around 1910 and was later called the Bandbox. The site is now the location of the William Fox building, right behind the Los Angeles Theatre.
See the listing for the Shamrock on the Theatres West of Broadway page under its later name, the Bandbox Theatre.