I’ve become interested in 3-manifolds with the following property (called ‘Property A’): let $c_{i}$ be a set of $g$ curves on a genus $g$ surface $Sigma$ and let $b_{i}$ be the $g$ meridional curves of $Sigma$. That is, if $Sigma$ is the boundary of a genus $g$ handlebody, then the $b_{i}$ bound compressing disks. Then ${c_{i}}_{i=1,dots,g}$ satisfy ‘Property A’ if, for any $b_{j}$, the geometric intersection numbers $iota(c_{i}, b_{j})$ have the same sign for all $i=1,dots,g$. Informally, all of the attaching curves for the 2-handles run over the 1-handles in the same direction.

So my question is, which 3-manifolds admit a Heegaard splitting with the attaching curves of the 2-handles satisfy Property A?

For example, any direct sum of lens spaces will satisfy this. As does the following Heegaard diagram for the PoincarĂ© homology sphere (taken from Manifold atlas http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_homology_sphere):

A possible nonexample could be the 3-torus, for which the only Heegaard diagram I know does not satisfy this property.

Lastly, and of course this depends heavily on my first question, is it possible that all rational homology spheres satisfy this?

I would be grateful for any sort of insight.