Dear Carolyn,
Yes, I too have heard that about anger. For me, the pain is
not anger, it is pain and tears under the anger.
The anger is just the defense against the pain. Once I can get past the anger and shame I feel for feeling angry, I experience the deep hurt in my heart. This hurt is the healing pain which softens the critical voice, and quiets the shaming voice allowing for a true release of suffering that quiets my angry spirit.
What I am trying to say is that I need to hurt, mourn, suffer the loss (let the tears flow)...which is pain. My anger is the first response, a defense...which then leads to the old hurt and the tears.
There is no need to be angry in life when others insult us - I can chose how to respond - I am just not there...... yet! 
We are supposed to take insults with quietness. But when my heart is still full of a lifetime defense of self-blame and victim anger it takes time to loosen that stuff up to get to the pain underneath -- that has been my healing path for the last many months.
Anger is the one emotion that others really retreat from and really have the hardest time dealing with it - I understand.
Anger and shame go hand in hand because as children, when we are being violated, our natural response is anger. But our caregivers will rarely allow for the expression of anger because it threatens them. The shame of our caregivers towards our anger stifles our voice, we have to stuff our anger along with the pain.
Some of us have more anger due to more victimization.
I hope this clarifies...it is not the anger that I am releasing...it is the raw hurt under the anger, does that make sense?
Just writing this out to you was a help for me.
Good question, thank you.
Lise