While teleparallel $f(T)$ gravity has shown considerable potential in addressing cosmological tensions, such as the $H_0$ and $S_8$ discrepancies, we explore previously overlooked solution spaces within this framework that hold further promise. Specifically, we examine the case where the customary assumption of a strictly positive effective dark energy (DE) density—natural in general relativity—may not apply, offering new possibilities. Focusing on the exponential infrared model $f(T)=T{e}^{T_{*}/T}$, where $T_{*}$ is a characteristic torsion scale, we investigate cosmological solutions parametrized by the dimensionless parameter $\beta =T_{*}/T_0$ with $T_0$ the present-day torsion scalar. This parameter uniquely determines the present-day matter density parameter ${\mathrm{\Omega }}_{\mathrm{m}0}$, and its sign plays a crucial role in characterizing deviations from the standard $\mathrm{\Lambda }$ cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$) expansion history. We elaborate on the structural asymmetry between the positive- and negative-$\beta$ branches: while the positive branch (${\beta }_{+}$) leads to dynamics with modest departures from $\mathrm{\Lambda }\mathrm{CDM}$, the negative branch (${\beta }_{-}$) yields more pronounced and nontrivial deviations at cosmological scales. We discuss that, despite these deviations, the negative-$\beta$ branch can remain consistent with local gravity constraints through an effective chameleonlike mechanism—wherein high-density environments naturally suppress deviations from the teleparallel equivalent of general relativity. We extend our analysis by examining the model in the context of dynamical DE. Ensuring consistency with cosmic microwave background (CMB) data, we find that the widely studied ${\beta }_{+}$ case exhibits phantom behavior, while the previously overlooked ${\beta }_{-}$ case—sufficient to avoid instabilities or ghosts—features a sign-changing DE density that transitions smoothly from negative to positive values at redshift $z_{†}\sim 1.5$, consistent with recent approaches to alleviating multiple cosmological tensions. Though the sign-changing DE density in the $f(T)$ model leads to a larger-than-expected enhancement, we further extend the analysis by incorporating a cosmological constant, $\mathrm{\Lambda }$. This extension, $f(T)\to f(T)+2\mathrm{\Lambda }$, broadens the solution space consistent with the SH0ES $H_0$ measurement while maintaining consistency with CMB power spectra. Additionally, it introduces richer phenomenological possibilities, including the potential moderation or cessation of cosmic acceleration at very low redshifts, aligning with recent observational analyses, such as those from DESI BAO data. Our findings also suggest that existing $f(T)$ models, as well as background-equivalent $f(Q)$ models, should be revisited in light of the novel theoretical insights presented here.

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• Alternative gravity theories
• Cosmology
• Dark energy