Case study: Dieteria asteroides

It has purple ray flowers, a yellow disk, and a whole series of phyllaries that make it look almost like a nettle viewed from the rear.

Key this out using Arizona Flora

Corollas bilabiate? No.
Flowers all perfect with strap shaped corollas? No.
Rays present? Yes.
Pappus of capillary bristles? Yes indeed! -- subkey F

I look ahead and see no Dieteria. No Dieteria in Arizona Flora. This species used to be Aster amplifolius according to SEInet. Of course AF has Aster, but no "amplifolius". Sigh ....

Following Ellen's suggestion to look up Taxonomy, I search Tropicos for Dieteria. Apparently this species was once called "Machaeranthera asteroides (Torr.) Greene". This genus is in AF (in the index at least, but the page references take us to "Aster"). Neither M. asteroides or A. asteroides is in Arizona Flora.

Whatever the case, let's continue with the key.

Rays white, purple, etc.  Yes!
Leaves and involucre with translucent oil glands? No.
Pappus of 1 or 2 awns? No, we have lots of bristles.
Ray achenes enveloped by subtending phyllaries? No.
Ray flowers have pappus? Indeed they do, just like disk.
Plants dwarf or otherwise? My plant was big and lanky.
Style tips lanceolate or rounded? Lanceolate.  -- Aster (27)

Leaves once or twice pinnatifid? No, my leaves are narrow and entire.
Stems low or taller?  Taller. (6)
Upper leaves tiny and scale-like? No. (9)
Leaves spiny toothed? No. (10)
Phyllaries glabrous or pubescent on back? Pubescent. (18)
Rays white or purple? Purple (19)
Leaves grasslike or toothed? Grasslike -- A. pauciflorus.

I look at Jim Verriers book (2018). He gives synonyms for D. asteroides, namely Aster tephrodes and Machaeranthera asteroides. A. tephrodes is in AF, so there is the link I am looking for.

Leaves grasslike or toothed? answer "not grasslike, usually toothed" (20)
Plants perennial, involucre not many ranked?  My involucre has lots of ranks. (22)
Herbaceous tips of phyllaries short or long? long -- and spreading. (24)
Phyllaries densely cinereous-puberulous, scarcely glandular.  Yes

Conclusions

Frank Roses book also has a similar looking species Machaeranthera tagetina (Spiny aster). SEInet indicates this was once "Aster tagetinus (Greene) Blake" and that is in AF. The key difference is the pinnatifid leaves, the first question in the Aster key.

D. asteroides is in the Jepson Manual. Here are some highlights from the description:

Biennial, Canescent-pubescent, often sparsely glandular. Phyllaries in 5-12 series. subspecies asteroides if the inflorescence is hemispheric (mine is I think).