  We give geometric formulae which enable us to detect (completely in some
cases) the regular homotopy class of an immersion with trivial normal bundle of
a closed oriented 3-manifold into 5-space. These are analogues of the geometric
formulae for the Smale invariants due to Ekholm and the second author. As a
corollary, we show that two embeddings into 5-space of a closed oriented
3-manifold with no 2-torsion in the second cohomology are regularly homotopic
if and only if they have Seifert surfaces with the same signature. We also show
that there exist two embeddings $F_0$ and $F_8 : T^3 \hookrightarrow {\bf R}^5$
of the 3-torus $T^3$ with the following properties: (1) $F_0 \sharp h$ is
regularly homotopic to $F_8$ for some immersion $h : S^3 \looparrowright {\bf
R}^5$, and (2) the immersion $h$ as above cannot be chosen from a regular
homotopy class containing an embedding.
