It cost close on £500 million to build, has a total floor area of 185,000 square metres and looks magnificent, although a bit like a giant food processor.
The tower, down which food might be stuffed in another incarnation, is 60 metres high and 100 metres across and conceals a fine cobbled courtyard, where the new European Parliament's first demonstration greeted the new arrivals.
But inside the huge glass construction, despite the light that floods the space-age multi-storey open concourses, there was anything but an atmosphere of calm.
Maps of the place only served to confuse, and members found that lifts only travelled to certain floors, escalators by-passed floors, and gangways crossing the central spine of the building could only be reached on certain levels.
Staff members have been suffering panic attacks as they struggle to find their way through the maze of corridors, and the new President of the Parliament admitted she had to climb nine flights of stairs to her office because the lifts were either not working or were likely to imprison her.
"I was afraid that if we got into the lift we might not have got out," she confessed.
One British Labour MEP compared his office to Cell Block H and said he expected to see prison guards escorting their charges around the courtyard.
In the members' dining-room there was chaos, with many MEPs, including Pat Cox, leaving hungry as staff admitted they had not bought enough food.
A furious Brian Crowley says he warned the architect 21/2 years ago about wheelchair access. In October he went to the chamber with officials and discussed the changes needed to get him to his group. Nothing happened.
To his amazement, promised changes to his office shower were being started only yesterday, and no one could explain why lifts would only go to the ground floor if the traveller had a special key.
"Security," he was told, despite the fact that escalators made the same trips possible for the able-bodied.
If the building was confusing to everyone, it was especially daunting for the 350 MEPs who had just been elected.
Dana Rosemary Scallon, the new independent member for Connacht-Ulster, said that at her meeting of new members (she has become an associate member of the group which includes Fine Gael) there was widespread anger that no one had been assigned to show them the ropes.
She kept the man from the Sun happy by telling him she had arranged for an evening of Irish traditional music in the Parliament.
Would she be singing? he asked. No, she said categorically.
But she admits that the years of recording and concert work in Germany and Holland have given her a useful recognition edge.
Like others, she has to decide which committees to opt for. She chose Regional Affairs as a priority, and the Youth, Education, Culture and Sport Committee as her second string.
She has a particular interest in the problems of music piracy on the Internet, and is likely to contribute to the debates on protecting intellectual property.
A learning curve? More like a precipice, she laughed.
