Apology of one maker of a slight mess:
Whether properties are front line or supporting depends very much on which war you are fighting. If you are interested in the linear locally convex theory then many properties are front line. But if you are mainly interested in smooth mappings, then quite few properties are front line; especially those which allow for uniform boundedness theorems (Greg should have added webbed in the diagram).
In fact, the space of smooth curves in a lcs (= locally convex space) does not change if you change the topology, as long as the bornology stays the same. In fact you can start from a dual pair of spaces $(E, E')$ (separating points on each other), check, whether Mackey Cauchy sequences for the weak topology converge (then $E$ is convenient), and then you can choose any lcs topology which is compatible with the duality. The finest such topology is bornological. This was part of the approach in the book of Froelicher and Kriegl, who started from scratch and reconstructed everything.
The most natural topology on $E$ from the point of view of Calculus is the final topology with respect to all smooth curves (equiv: all Mackey convergent sequences, equiv: all locally Lipschitz curves, ...); this denoted $c^\infty E$. The finest lcs topology coarser than $c^\infty E$ is the bornologification of any lcs toplogy which is compatible with dual pair.
Or you can try to allow users to make use of knowledge in the theory of lcs. This was the approach in the book: Convenient setting ... There you are allowed to use any lcs topology on $E$ that you know well or can describe well and which still has the given bornology. Spaces $E$ are identified if they are biboundedly linearly isomorphic (equiv.: diffeomorphic). To be convenient is a property of such equivalence classes of spaces.
So, in Greg's nice diagram above, the place of convenient should be: Sequentially complete $\implies$ convenient.
In fact, each "naturally described" lcs is convenient. You have to force it to be not convenient by choosing a not Mackey complete subspace with the induced lcs topology.