This paper studies so-called twisted-austere pairs, given by (M,μ) with Mk⊂Rn a Riemannian submanifold of dimension k, μ a 1-form on M, and such that the subbundle N∗M+μ⊂T∗Rn≅Cn is a special Lagrangian submanifold. This condition imposes strong constraints on the geometry of the submanifold M. More precisely: for any normal direction ν to M, the second fundamental form Aν of M along ν, and the 1-form μ have to satisfy a certain system of coupled non-linear PDE’s, called the twisted-austere equations. The case μ=0 is well known, and the corresponding submanifolds M are called austere; this case is characterized by the condition that the spectrum σ(Aν) has to satisfy −σ(Aν)=σ(Aν) as a set. In particular, austere submanifolds are minimal. After reviewing the PDE’s of twisted-austere pairs (M,μ), a complete classification of these pairs is given firstly for the easy case when M is totally geodesic, secondly for dimensions k=1,2, and thirdly for dimension k=3. For dimension k=1 it is proved that M is a straight line, and for k=2 it is shown that M has to be a minimal surface and μ an harmonic 1-form on M. The case k=3 is the bulk of the paper, and its classification consists of a careful analysis of the twisted-austere equations. The main conclusion (Theorem 3.1.) is that either M3⊂Rn is ruled by lines, or n=5 and M3⊂R5 is a generalized helicoid ruled by planes.