I touched on this in an earlier post, but I'll repost it here

I would say overall it's very accurate. Other than a few exceptions, they were incredibly loyal to the throne, even if the occupant of the throne wasn't the person they were supposed to protect.

To begin with, they had very good reason to be loyal. They were better paid than normal soldiers and were given the unique right of, on the occasion of the natural death of an emperor, being able to raid the imperial treasury and each man was allowed to take as much gold and treasure as they could carry, often making them very rich. It should be noted, however, that this is disputed. I don't know of any secondary sources other than Blondal and Benedikz that bring it up, and I would tend to agree it's probably not quite how it actually happened. It's unsubstantiated, but I would guess that the varangians were allowed to go into the treasury and take treasure but were either limited to how much they could take or the leaders were allowed to go in but were forced to distribute it among their men. Furthermore, much of the Varangian guard was based outside of Constantinople, so it's possible that even if the practice was as is described, it was limited to those where in the city (which, to my understanding, were those who had gained recognition). The money alone was usually enough to keep their loyalty, but the prestige they got was another. They were almost always used as the elite heavy infantry in a battle and were often with the Emperor himself during battle, so they developed a relationship with the man they were supposed to protect. Furthermore, they were commanded by a Greek commander who the Emperor would appoint personally, so in theory they were commanded by a man who was loyal.

Their loyalty, or I should say their desire for pay, was heavily tied to there being someone on the throne who would actually pay them, so the Varangians were willing to put their support to whoever was around to do that. This is why on the death of Nikephoros II they immediately bowed down to John I, he was in charge now so he was the guy who was going to pay. It's because of this loyalty to their pay that there are a few times it seems that their loyalty wavered. You mentioned Michael V, but it's worth mentioning that Michael V may have been the stupidest person to ever don the purple in the history of Byzantium. In a few short months, he managed to alienate everyone against him by exiling the beloved heirs to the royal dynasty, Zoe and Theodora, he turned the patriarch against him, and as such the people of Constantinople were completely against him. With the Varangians, there were rumors about that he was planning on replacing them with Scythian slaves. This meant that they stood to lose their prominence, and as such their pay, and so they were quick to throw their support to the people who would pay them, namely Zoe and Theodora. Another situation that was sort of similar to this was the deposition of Romanos IV in favor of Michael VII. It should be noted that in this situation it was proclaimed by John Doukas that Romanos IV was dead, and John was also rallying the support of the remaining armies to ensure that even if he wasn't, he wasn't going to reclaim his throne. The varangains knew they stood no chance oh gaining from continuing to support Diogenes (although it should be mentioned that at Manzikert, many of them were slaughtered defending their patron, so the majority of them really did fight to the bitter end), and so they threw their support behind the guy who was actually doing the Emperor-ing at this point. Really, the only time I can find of just straight up unjustified disloyalty is their mutiny against Nikephoros III when he wanted to blind Nikephoros Byrennios. I unfortunately don't have sources on hand to quite explain this, but I think it's worth mentioning that in such a long history of a body guard, only having such a small amount of mutinies is a pretty stellar track record. It's because of this that the Emperors were very content to let them protect them, because unless something absolutely disastrous happened, the Varangians were their with their axes to ensure that they didn't die.

Source) Blöndal and Benedikz' The Varangians of Byzantium.