Unfortunately the most recent review specialized on this topic (I found) dates back to 2004, and it's pretty confusingly written, IMHO. The general gist of it appears to be that T cells can act as APC cells under some conditions (aka T-APC) and it was generally thought that this serves a down-regulatory function for T cells, e.g.
There is considerable evidence, dating as far back as
1978, in support of the notion that human activated
T cells can function as APC (so-called T-APC) and
thereby affect the activities of other T cells. [...]
Most reports show that MHC class II–mediated antigen
presentation by T-APCs provides downregulatory signals
by inducing either apoptosis or clonal anergy in activated
T cells or cytotoxicity in resting T cells.
This T-T presentation-induced anergy does not appear to
result from TCR ligation in the absence of costimulatory
molecules, because activated T cells themselves express
CD80 or CD86.
More recent work has found that trogocytosis is possible between T cells, i.e. they can pass each other peptide-bound MHC. In particular trogocytosis explains some mysteries in this area, such as how mouse CD8 T cells, which cannot express the MHC-II gene, themselves actually acquire such receptors on their surface.
One such recent paper proposes that such T-T interactions differentiate CD4 T cells into regulatory ones vs Th17 pro-inflammatory ones:
We investigate the functional outcome of T-T cell antigen presentation by CD4 T cells and find that the antigen-presenting T cells (Tpres) predominantly differentiate into regulatory T cells (Treg), whereas T cells that have been stimulated by Tpres cells predominantly differentiate into Th17 pro-inflammatory cells.
The introductory section of this 2021 paper contains a useful minireview of the directions of research (some of them being apparently somewhat contradictory to each other, if more specific conditions of the experiments aren't taken into account)
Another important mechanism of intercellular communication within the immune system is trogocytosis (Joly and Hudrisier, 2003), where T cells and natural killer (NK) cells acquire micrometer-sized fragments of the APC membrane (Carlin et al., 2001; Vanherberghen et al., 2004). These fragments can contain immune-modulating membrane receptors such as killer inhibitory receptors and major histocompatibility complex (MHC) class I and II complexes. Previously it was shown that MHC-II vesicles are released by APCs and acquired by responding T cells (Arnold and Mannie, 1999; Patel et al., 1999). T cells also acquire peptide MHC (pMHC) complexes at the IS [immunological synapse] via a process that requires T cell receptor (TCR) internalization and TCR signaling (Huang et al., 1999; Hudrisier et al., 2001). In addition, they acquire CD80 and CD86, ligands of the co-stimulatory receptor CD28 (Hwang et al., 2000; Sabzevari et al., 2001). This could convert T cells into APCs. Both CD8 T cell-dendritic cell (DC) interactions and secondary T cell-T cell interactions are required to generate protective CD8 T cells (Gérard et al., 2013), and CD8 recall responses depend on CD8 T cells taking up MHC-II from DCs to present it to CD4 T cells and obtain their help (Romagnoli et al., 2013). In contrast, other results have suggested that pMHC acquisition by T cells may play an inhibitory role (Dhainaut and Moser, 2014). For instance, T cell antigen presentation to other T cells has been associated with the induction of anergy, apoptosis, and even tolerance (Chai et al., 1998; Mannie et al., 1996), and may represent a mechanism to limit clonal expansion (Tsang et al., 2003).