When the ordinal sum equals the Hessenberg ("natural") sum Let $\alpha_1 \geq \ldots \geq \alpha_n$ be ordinal numbers.  I am interested in necessary and sufficient conditions for the ordinal sum $\alpha_1 + \ldots + \alpha_n$ to be equal to the Hessenberg sum $\alpha_1 \oplus \ldots \alpha_n$, most quickly defined by collecting all the terms $\omega^{\gamma_i}$ of the Cantor normal forms of the $\alpha_i$'s and adding them in decreasing order.
Unless I am very much mistaken the answer is the following: for all $1 \leq i \leq n-1$, the smallest exponent $\gamma$ of a term $\omega^{\gamma}$ appearing in the Cantor normal form of $\alpha_i$ must be at least as large as the greatest exponent $\gamma'$ of a term $\omega^{\gamma'}$ appearing in the Cantor normal form of $\alpha_{i+1}$.  And this holds just because if $\gamma' < \gamma$, 
$\omega^{\gamma'} + \omega^{\gamma} = \omega^{\gamma} < \omega^{\gamma} + \omega^{\gamma'} = \omega^{\gamma'} \oplus \omega^{\gamma}$.  
Nevertheless I ask the question because:
1) I want reassurance of this: I have essentially no experience with ordinal arithmetic.
2) Ideally I'd like to be able to cite a standard paper or text in which this result appears.  
Bonus points if there happens to be a standard name for sequences of ordinals with this property: if I had to name it I would choose something like unlaced or nonoverlapping.
P.S.: The condition certainly holds if each $\alpha_i$ is of the form $\omega^{\gamma} + \ldots + \omega^{\gamma}$.  Is there a name for such ordinals?

You are exactly right about the "asymmetrically absorptive" nature of standard ordinal addition (specifically with regard to Cantor normal form). Your condition is necessary and sufficient (sufficiency is easy, and you've shown necessity). I don't know of any standard name for such sequences, though. As for your $\omega^\gamma+\cdots+\omega^\gamma$ bit, that isn't appropriate for Cantor normal form. We require the exponents to be listed in strictly decreasing order.
