BroadwayWorld noted that the email further said, in bold, underlined lettering, "If you are not full time, you do not have to sign up to do this work. If you are full time, you are obligated."
But in an email statement Friday, the Madison Square Garden Co. said, "The Radio City Rockettes are proud to participate in the 58th Presidential Inaugural. For a Rockette to be considered for an event, they must voluntarily sign up and are never told they have to perform at a particular event, including the inaugural."
"It is always their choice," the company said. "In fact, for the coming inauguration, we had more Rockettes request to participate than we have slots available. We eagerly await the inaugural celebrations."
Hours after MSG issued that statement, amid a stinging backlash to the union's stance, he American Guild of Variety Artists issued a statement to the press that was now supportive of its members, instead of scolding.
"MSG's announcement that the Rockettes were being included in the Presidential inaugural has brought up legitimate concern among our Members, the theatrical community and the public at large," the union said..
"We took this very seriously and immediately contacted Radio City for a meeting to address this volatile situation. This is always the first course of action in a labor-management issue or dispute," the union said. "Fortunately, the Company has agreed that ALL participation in this particular event will be voluntary. We are greatly relieved and hope to work with our Members to inform them and alleviate the anxiety and fears that this has caused."
"There has also been a considerable amount of misinformation about the Union's position. The Union never 'demanded' that the Rockettes perform at the inauguration. A message was sent to the Rockettes last evening that stated the terms of their contract," the union said.
"There is a small group of year round Rockettes who are contractually obligated to perform at scheduled events throughout the year. We are pleased that Radio City has agreed that for those Rockettes with year-round employment,participation in this event will be voluntary as well."
"AGVA is proud of its long history of rejecting sexism, racism, homophobia and all forms of discrimination and harassment in the workplace," the union said.
CNBC has reached out for comment on the Rockettes controversy to Trump's transition team.
In addition to the Rockettes, the Mormon Tabernacle Choir is also scheduled to perform for Trump's inaugural festivities. So is Jackie Evancho, a 16-year-old who has appeared on the TV show "America's Got Talent."
Earlier this week, the show-business trade news site TheWrap reported that Trump was so unhappy about the difficulty his team has had wrangling A-list performers for the inaugural events that "he's ordered a 'Hail Mary' shake-up of his recruiters to try to book performers."
"Mark Burnett, who produces 'Celebrity Apprentice' with Trump and has been overseeing entertainment for inauguration festivities, has brought in talent recruiter Suzanne Bender, a former 'Dancing With the Stars' and 'American Idol' booker, to end an effective freeze-out by Hollywood," TheWrap reported, quoting a Trump insider.
The move came after a number of celebrities reportedly nixed the idea of performing for Trump, including Celine Dion, Elton John, Garth Brooks, as well as Andrea Bocelli and the rock group Kiss.
Trump's inaugural chair has denied reports that Bocelli turned down Trump, saying that it was the tenor who offered to perform but was declined by Trump.
Trump seemingly responded to those reports on Thursday with an angry tweet that suggested celebrities were the ones pursuing his attention, instead of the other way around.
— CNBC's Michelle Castillo contributed to this report.