The structures of 15 La-Al-Si-O glasses, whose compositions span 11-28 mol% La2O3, 11-30 mol% Al2O3, and 45-78 mol% SiO2, are explored over both short and intermediate length-scales by using a combination of solid-state Al-27 magic-angle spinning (MAS) nuclear magnetic resonance (NMR) spectroscopy and molecular dynamics (MD) simulations. MAS NMR reveals Al speciations dominated by AlO4 groups, with minor but significant fractions of AlO5 (5-10%) and AlO6 (less than or similar to 3%) polyhedra present in all La2O3-Al2O3-SiO2 glasses; the amounts of Al-[5] and Al-[6] coordinations increase for decreasing molar fraction of Si. The MD simulations reproduce this compositional trend, with the fractional populations of AlOp groups (p = 4, 5, 6) according well with the experimental results. The modeled La speciations mainly involve LaO6 and LaO7 polyhedra, giving a range of average La3+ coordination numbers between 6.0 and 6.6; the latter increases slightly for decreasing Si content of the sample. Besides the expected bridging and non-bridging O species, minor contributions of oxygen triclusters (<= 9%) and free O-2(-) ions (<= 4%) are observed in all MD data. The glass structures exhibit a pronounced Al/Si disorder; the MD simulations reveal essentially random SiO4-SiO4, SiO4-AlOp and AlOp-AlOq (p, q = 4, 5, 6) associations, including significant amounts of AlO4-AlO4 contacts, regardless of the n(Al)/n(Si) molar ratio of the glass. The strong violation of Al-[4]-Al-[4] avoidance is verified by 2D Al-27 NMR experimentation that correlates double-quantum and single-quantum coherences, here applied for the first time to aluminosilicate glasses, and evidencing AlOp-AlOq connectivities dominated by AlO4-AlO4 and AlO4-AlO5 pairs. The potential bearings from distinct fictive temperatures of the experimental and modeled glass structures are discussed.