Introduction:Small bodies in the trans-Neptunian region are key to understanding Solar System formation and evolution. Trans-Neptunian objects (TNOs) are relics of planetary formation in the outer protoplanetary disk, but most of them were later affected by the giant planet instability and the dynamical evolution of the Solar System, with the remarkable exception of the so-called "Cold Classical" population.TNOs incorporated ices and other solids from the protosolar disk and thus provide precious information about early conditions in the disk. However, many TNOs also underwent evolutionary processes such as melting, differentiation, segregation, fragmentation, and irradiation that modified the original protoplanetary composition. In this work, we focus on mid-sized objects (diameter between about 50 and 1000 km), representing a generation of outer planetesimals that suffered limited differentiation and collisional evolution. Methods:We used the low spectral resolution PRISM grating on the Near-Infrared Spectrograph (NIRSpec, 0.7-5 µm) of the James Webb Space Telescope (JWST) to observe 75 medium-sized TNOs observed in several programs: the year 1 Guaranteed Time Observations (GTO) Program "Kuiper Belt Science with JWST" (GTO-KBO, ID1191, ID1231, ID1272, and ID1273), the Cycle 1 Large Program ID2418 "Discovering the Surface Composition of trans-Neptunian objects" (DiSCo-TNOs), the Cycle 2 Program ID3991 "Small Cold Classical TNOs as Witnesses of Outer Nebular Chemistry", and the Cycle 3 Program ID4665 "Constraining the origin and dynamical evolution of extreme trans-Neptunian objects through NIR spectroscopy". The objects within the sample span the diversity of the TNO population (excluding the volatile-rich dwarf planets and the Haumea family) in terms of size, visible colors, geometric albedo, and dynamical properties.For a larger comparison, we also include 10 Centaurs observed in the DiSCo-TNOs and GTO Programs, and 8 Neptune Trojans observed within the Cycle 1 Program ID2550 "The First Near-IR Spectroscopic Survey of Neptune Trojans".Following a similar approach to the one that we successfully used to analyze the DiSCo-TNOs data [1,2], we analyzed the extended set of spectra with different clustering techniques (Principal Component Analysis, k-means, hierarchical clustering) to highlight the spectral diversity of the targets. We obtained information about the icy molecular composition by identifying several bands of interests and by calculating their band areas and positions. To date, this represents the largest near-IR spectral dataset of icy bodies, providing the most complete picture of the molecular composition of outer Solar System planetesimals. Results:While the new objects have very diverse sizes and orbital parameters, the vast majority of their spectra fall into the three main compositional categories identified by DiSCo: the so-called "Bowl" surfaces, which are dominated by water ice features and are also dust-rich and CH-poor; the "Cliff" and "Double-dip" surfaces, which are carbon-rich and water-poor, with Double-dip being particularly rich in CO2 and CO, and Cliff surfaces being rich in organics. We confirm the detection of several icy molecules, including H2O, CO2, 13CO2, CO, CH3OH, and complex molecules and refractory materials containing aliphatic C-H, C≡N, O-H, and N-H bonds. The band areas of the different molecules, sensitive to both abundance and path length in solid-state ices, vary significantly among different icy bodies and correlate with the identified spectral categories.In addition, thanks to the larger sample, at least three sub-categories of Bowl-TNOs and three sub-categories of Cliff-TNOs can be identified. The surfaces of the Cliff1 sub-group are ice-rich, with prominent CH3OH, H2O, and CO2 ice features that are much weaker in the ice-poor Cliff2 sub-group [3]. The Cliff1-CO2 sub-group includes transitional objects that resemble Cliff1 surfaces, but exhibit very strong CO2 features, similar to the CO2 spectral properties (position and area) of Double-Dip TNOs. Finally, two TNOs and two Neptune Trojans are unclassified, showing weak icy features that are close to those observed in "shallow"-type Centaurs [2], possibly due to previous episodes of ice sublimation. Except for the rare Cliff1-CO2 TNOs, very sharp transitions are observed between the different spectral groups.Cold Classical TNOs belong almost entirely to the Cliff2 sub-group, while the other dynamical classes of icy bodies (Scattering Disk Objects, Resonant Objects, Hot Classicals, Detached and Extreme Objects, Neptune Trojans, Centaurs) exhibit variable proportions of the different spectral categories, with no statistically significant association between dynamical classes and any specific spectral category. Discussion:Generally speaking, most TNO surfaces show significant deviations from the protoplanetary and cometary ice compositions, revealing that specific evolutionary processes shaped the molecular composition in the outer Solar System before or just after the planetesimals' formation. The fact that, except for the Cold Classicals, the different dynamical classes show variable amounts of the spectral categories suggests that late evolutionary processes, such as prolonged exposure to the space environment and irradiation, are not the main drivers in shaping the spectral groups. So far, the only clear irradiation trend observed is in non-Cold Classical Cliff1-TNOs, whose methanol bands decrease with increasing residence outside the heliosphere, where cosmic ion fluxes are higher. [3].An early sculpting is necessary to create the distinct separation of the spectral clusters. In particular, a sharp process, such as the one associated with ancient icelines, must be invoked to explain the significant variations observed in the surface molecular constituents. The currently favored scenarios include either the pre-accretional CO iceline on grains in the protoplanetary disk, and/or the post-accretional retention icelines of CO2 and CH3OH at the surface of planetesimals just before a major planetary migration. In both cases, planetesimals probably formed in this order of increasing distance from the Sun: Bowl<Double-Dip<Cliff1<Cliff2.Finally, we observe a significant and intriguing lack of CO2-rich objects for perihelion distances smaller than about 31 AU. We explore two different scenarios to explain this dichotomy, the first one due to physicochemical processes of CO2 loss, and the second one related to dynamical processes of preferential injection of Bowl-type TNOs from the inner Oort-Cloud to the Centaurs region. [1] Pinilla-Alonso N., et al., 2025, NatAs, 9, 230. doi:10.1038/s41550-024-02433-2[2] Licandro J., et al., 2025, NatAs, 9, 245. doi:10.1038/s41550-024-02417-2[3] Brunetto R., et al., 2025, ApJL, 982, L8. doi:10.3847/2041-8213/adb977