Having recently worked on this topic, I can say that the answer to both of your questions is yes.
The belief states may be partially ordered for example applying the monotone likelihood ratio, or other appropriate stochastic orderings that tell when a probability mass function (pmf) $f$ -- here a belief state -- is "greater than" another pmf $g$.
Roughly speaking, if the POMDP state transition and observation models are such that the partial order is preserved on a belief update operation, structural statements about the optimal policy of the POMDP can be made. Details may be found in a recent journal paper: V. Krishnamurthy, U. Pareek, Myopic Bounds for Optimal Policies of POMDPs: An extension of Lovejoy’s Structural Results, Operations Research, 2014; available at the author's website. The article also provides a literature review on the topic where further references may be found.
Krishnamurthy and collaborators also have a paper where the MLR ordering is applied on a single line in the belief space, on which the MLR order applies to every belief state: V. Krishnamurthy, D. Djonin, Structured Threshold Policies for Dynamic Sensor Scheduling--A POMDP Approach, IEEE Trans Signal Processing, Vol.55, No.10, pp.4938--4957.