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This paper reports the first measurement of the transverse momentum (𝑝T) spectra of primary charged pions, kaons, (anti)protons, and unidentified particles as a function of the charged-particle flattenicity in pp collisions at √𝑠 =13  TeV. Flattenicity is a novel event shape observable that is measured in the pseudorapidity intervals covered by the V0 detector, 2.8 <𝜂 <5.1 and −3.7 <𝜂 <−1.7. According to QCD-inspired phenomenological models, it shows sensitivity to multiparton interactions and is less affected by biases toward larger 𝑝T due to local multiplicity fluctuations in the V0 acceptance than multiplicity. The analysis is performed in minimum-bias (MB) as well as in high-multiplicity events up to 𝑝T=20  GeV/⁢𝑐. The event selection requires at least one charged particle produced in the pseudorapidity interval |𝜂| <1. The measured 𝑝T distributions, average 𝑝T, kaon-to-pion and proton-to-pion particle ratios, presented in this paper, are compared to model calculations using pythia 8 based on color strings and EPOS LHC. The modification of the 𝑝T-spectral shapes in low-flattenicity events that have large event activity with respect to those measured in MB events develops a pronounced peak at intermediate 𝑝T (2 <𝑝T <8  GeV/𝑐), and approaches the vicinity of unity at higher 𝑝T. The results are qualitatively described by pythia, and they show different behavior than those measured as a function of charged-particle multiplicity based on the V0M estimator.