My knowledge of the comics is spotty at best (they've been going for over 50 years after all), but I can attempt to answer this from a perspective rooted in the Marvel Cinematic Universe. I do not know if this will match the way that The Hulk is seen in the comics or not. This answer will contain spoilers for both The Avengers/Avengers Assemble and Avengers: Age of Ultron.
Bruce Banners transformation into The Hulk within the Marvel Cinematic Universe has been triggered in multiple ways, the most reliable and frequent being injury, anger or extreme stress. In short, we have only ever seen Bruce Baner turn into The Hulk when his fight or flight response is triggered.
The Hulk as an entity is near invulnerable to damage, incredibly strong and more than capable (and willing) to handle himself in a fight. In the Marvel Cinematic Universe, it is worth noting that The Hulk can be directed or distracted - as if his target is whatever he perceives to be the greatest threat to him at the time. We see this in action in The Avengers/Avengers Assemble during the Helicarrier scene - his intial target is Black Widow, who was threatening him when he picked up the Scepter just moments earlier, but he switches targets to Thor and then a fighter plane whenever a new target seems to be a larger threat. His ability to prioritise threats and deal with them appropriately is the ideal action when the fight or flight response is activated.
We see in multiple occasions that between the time of the threat being neutralized and his transformation back into Bruce Banner, The Hulk is often nearly as calm as those around him, again showcased in The Avengers/Avengers Assemble when he is able to catch Iron Man from falling and thereby saving his life. A few moments later, he accompanies the rest of The Avengers to go and confront the defeated Loki - and he does this without attacking him. Likewise, at the end of Avengers: Age of Ultron we see him quite calmly sitting on a Quinjet, seemingly unhappy or regretful - but not angry.
Which leads me to believe that The Hulk as an entity isn't as angry as everyone thinks. Bruce Banner's transformation into The Hulk may be triggered by anger (amongst other things) and that emotion carries over into The Hulk, but left to his own devices we can clearly see that he will calm down after a period. Unfortunatley, his initial anger can cause a looping effect when people begin to see him as a threat and attack him, making him angrier and more destructive, which makes more people attack him which makes him angrier and more destructive which makes more people attack him...
In short, throughout the Marvel Cinematic Universe we have been shown that The Hulk is no less capable of calmness than anyone else - it's just that when The Hulk does show up, it's often during a time when anger is a very appropriate response, which can in turn make him angrier.
It's probably also worth mentioning that in the comics I have read (such as Planet Hulk), The Hulk is again just as capable of not being angry as any else would be.