The relationships between the three main clades of Pterygota – Odonata, Ephemeroptera and Neoptera – are currently unresolved. Three hypotheses have been proposed:[1][2][3]
I haven't found a source explaining what's going on here yet, but there seems to be two different definitions for this group:
Palaeoptera + Polyneoptera + Paraneoptera, i.e. all winged insects where wings develop outside the body (sources include [1]). This was also the original definition used by Sharp, 1898, except Sharp placed Mallophaga and Anoplura (chewing and sucking lice respectively) instead in "Anapterygota" (secondarily wingless insects), which also included Siphonaptera (fleas).
Polyneoptera + Paraneoptera, i.e. A division of Neoptera where wings develop outside the body.
To put it in other words, sometimes Odonata (dragonflies and damselflies) and Ephemeroptera (mayflies) are included in Exopterygota, and sometimes they are not...
has incomplete metamorphosis (Polyneoptera was formerly part of Exopterygota)
evidence according to Kjer et al. (2016): tegmina, enlarged anal field (vannus) of hind wing [see Insect wing#Fields], euplantulae
'Kristensen's comb' refers to the polytomy of Kristensen (1981), in which a polytomy is given for the "lower Neoptera" (= most of Polyneoptera, with Zoraptera placed in Paraneoptera)
Proposed clades in Polyneoptera:
Dictyoptera (= Blattodea [including Isoptera] + Mantodea] [supported in consensus according to Kjer et al. (2016)]
Extinct group of insects from the Palaeozoic. Historically their position and composition was controversial, but recent studies have resolved it as the sister group to Dictyoptera.[1][2][3]
Alternatively it has been placed in its own superorder Paoliidea in infraclass Gryllones (= Polyneoptera).[4] (I'm assuming this isn't widely accepted, given the use of a different (Russian?) insect classification and nomenclature based on Rasnitsyn & Quicke (2002)'s History of Insects, where e.g. Scarabaeona = Pterygota, Scarabaeones = all winged insects except Gryllones (= Neoptera + Palaeoptera).)
^Jakub Prokop; Wieslaw Krzemiński; Ewa Krzemińska; Thomas Hörnschemeyer; Jan-Michael Ilger; Carsten Brauckmann; Philippe Grandcolas; André Nel (2014). "Late Palaeozoic Paoliida is the sister group of Dictyoptera (Insecta: Neoptera)". Journal of Systematic Palaeontology. 12 (5): 601–622. doi:10.1080/14772019.2013.823468. S2CID84407734.
^Sroka, P.; Staniczek, A. H.; Bechly, G. (2015). "Revision of the giant pterygote insect Bojophlebia prokopi Kukalova-Peck, 1985 (Hydropalaeoptera: Bojophlebiidae) from the Carboniferous of the Czech Republic, with the first cladistic analysis of fossil palaeopterous insects". Journal of Systematic Palaeontology. 13 (11): 963–982. doi:10.1080/14772019.2014.987958. S2CID84037275.
(Grey notes are my attempts to match the extant families in Bechly's classification with the equivalent families/subfamilies/etc under the World Odonata List classification)