η1 or sigma complex involving cyclopentadienyl anion
Even though depictions of $\eta^1$-coordination might suggest a $\mathrm{sp^3}$-carbon donating, there is little evidence, that it actually is. Taking the example you have presented in your answer from Casey et.&nbsp;al.[1] I have run a geometry optimisation to obtain molecular orbitals at the DF-BP86/def2-SVP level of theory. The following geometry parameters around the coordinating carbon already suggest only little pyramidalisation:
\begin{array}{lrr}\hline
\text{Parameter} &amp; \text{Experimental [1]} &amp; \text{DF-BP86/def2-SVP}\\\hline
\mathbf{d}(\ce{Re-C^1}) &amp; 236.0 &amp; 240.2\\
\mathbf{d}(\ce{Re-H})   &amp; 288.7 &amp; 282.6\\
\mathbf{d}(\ce{Re-C^2}) &amp; 317.8 &amp; 321.3\\
\mathbf{d}(\ce{Re-C^5}) &amp; 315.6 &amp; 315.6\\
\angle(\ce{Re-C^1-H})   &amp; 114.3 &amp; 100.7\\
\angle(\ce{Re-C^1-C^2}) &amp; 110.8 &amp; 109.5\\
\angle(\ce{Re-C^1-C^5}) &amp; 108.5 &amp; 106.4\\
\angle(\triangle[\ce{C^1C^2C^5}]\ce{-H}) &amp;
                        122.8   &amp; 139.1\\
\text{RMSD} &amp; 0.0 &amp; 0.1\\\hline
\end{array}

We know that crystal structures are not really that accurate with protons, hence I'd put a little more trust into the calculation at this point. The paper does not give a value for any protons.
Here is a picture of the calculated structure.



When we look at the canonical MO we will see that all (occupied) &pi;-obitals of the Cp-ligand are still there. Obviously they are distorted, but clearly recognisable. 



The second orbital is the HOMO-1, which is the main contribution for the coordination, and the last is the HOMO (click for large). The only s-contribution to the HOMO-1 is from the carbon of the $trans\text{-}\ce{CO}$-ligand (and some hydrogen).
In output numbers, missing from 1 are minor contributions (&lt;0.05):


 Alpha occ 81 OE=-0.276 is C39-p=0.19 C42-p=0.12 C41-p=0.12 C43-p=0.11 C40-p=0.11
 Alpha occ 87 OE=-0.185 is C39-p=0.26 C42-p=0.23 C41-p=0.22 Re24-d=0.08 C27-s=0.05
 Alpha occ 88 OE=-0.153 is C43-p=0.32 C40-p=0.32 C41-p=0.13 C42-p=0.12
 

When we localise the orbitals with the Natural Bond Orbital (NBO) analysis, we find a lone-pair at $\ce{c^1}$ (blue/orange) which donates into the anti-bonding $\ce{Re-^{$trans$}CO}$ orbital (yellow/red):


The composition of the lone pair orbital is


     (Occupancy)   Bond orbital / Coefficients / Hybrids
 36. (1.27157) LP ( 1) C 39            s(  8.77%)p10.40( 91.23%)d 0.00(  0.00%)


The low occupancy is normal for donor-acceptor-complexes with some covalent character.

TL;DR Most $\eta^1$ complexes are in first approximation &pi;-coordination complexes like the definition suggests[2]. This may change significantly when moieties are altered.



Further thoughts:
Wikipedia makes one important statement (it rarely does, but here it is true)3:


  Molecules with polyhapto ligands are often fluxional, also known as stereochemically non-rigid. 


You can expect that in $\eta^1$-ligands the metal can easily migrate from one carbon to another, as the MO show. I would expect some kind of pseudo-rotation.

Ref.


Charles P. Casey, Joseph M. O'Connor, William D. Jones, Kenneth J. Haller. Organometallics 1983, 2 (4), 535–538. (link via doi, pdf via researchgate.net) Crystallographic data via CCDC number: 1118829.
η (eta or hapto) in inorganic nomenclature
Wikipedia: Hapticity and fluxionality