Let $E\xrightarrow{p} \Sigma X$ be a principal G-bundle over a suspension. Write $\Sigma X= C\_+X\cup\_X C\_-X$. Then there are trivialisations of the restrictions $E|\_{C\_+X}\cong C\_+X\times G$, $E|\_{C\_+X}\cong C\_-X\times G$, and the transition function between them over their intersection X is defined by a map $t:X\rightarrow G$. The homotopy class of this map completely characterises the bundle, and the process is in fact reversible. Given such a clutching function, one can construct a unique bundle over the suspension.


So if $f:\Sigma X\rightarrow BG$ is a map classifying the G-bundle E, how does this map relate to the clutching function $t:X\rightarrow G\simeq \Omega BG$ ? How does one go between one and the other?


The adjoint isomorphism $[\Sigma X, BG] = [X,\Omega BG]= [X,G]$ obviously has a part to play, but for all I have tried I cannot for the life of me craft a strong enough argument to really convince myself of their true relationship.



Clutching functions and Classifying maps
Here is one possible way of answering the question, using the simplicial model for $EG$. In this context, we can show very explicitly that the map $\Sigma X\to BG$ built (via adjointness) from the clutching function for $E$ classifies $E$ (and then if $X$ is a CW complex, any other classifying map for $E$ is homotopic to this one, by a standard cell-by-cell argument).


If $E\to \Sigma X$ is a principal $G$-bundle, then as explained in the question, $E$ is trivial over each half of $\Sigma X$. So $E$ is (isomorphic to) the quotient space 
$$(C\_{-} X\times G) \cup (C\_+ X \times G)\,/\,(x\_-, g)\sim(x\_+, c(x)g)$$
where $c: X\to G$ is the clutching function, and the notation for the relation is supposed to indicate that $x\_-$ and $x\_+$ are the copies of $x$ in the bases of the cones $C\_-X$ and $C\_+ X$ (respectively). 


Let's assume that $G$ is a Lie group, so that $BG$ can be taken to mean the bar construction on $G$ (as in, say, "Segal's paper Classifying spaces and spectral sequences"). Then there is a natural map $\Sigma G \to BG$ given by the fact that $G$ is the space of 1-simplices in $BG$, and it has an adjoint $i: G\to \Omega BG$ (which sends $G$ to the loop that "runs around the 1-simplex labeled '$g$'," so to speak).


The function $i \circ c: X \to \Omega BG$ has an adjoint $f: \Sigma X \to BG$. It sends $x$ to $[c(x), t]$, which means "the point t units along the 1-simplex labeled $c(x)$." We want to prove that $f$ classifies $E$ - that is, we want to construct a map $\tilde{f}: E\to EG$ that is $G$-equivariant and covers $f$. 


This can be done very explicitly. In Segal's setup, $EG$ is also a simplicial space, with 1-simplices parametrized by $G\times G$; the $G$-action on this (universal) principal bundle is induced by right-multiplication in $G$, and the map $EG\to BG$ sends the 1-simplex $(g,h)$ to $g h^{-1}$. 


I want to think of $\Sigma X$ as $X\times I$ with both ends collapsed to points, so I'll take $C\_-$ to be parametrized by $t\in [0,1/2]$ and $C\_+$ to be parametrized by $t\in [1/2, 1]$.


We can define $\tilde{f}$ separately on the two halves of $E$: on the left half 
$$C\_{-} X\times G = \left[(X\times [0,1/2])/(X\times 0)\right]\times G,$$
 we set $\tilde{f}(x, t, g) = [(c(x)g, g), t]$. On the right half 
$$C\_{+} X\times G = \left[(X\times [1/2, 1])/(X\times 1)\right]\times G,$$ 
we set $\tilde{f}(x, t, g) = [(g, c(x)^{-1}g), t]$. Note that in both cases, when we project down to $BG$ we get just $[c(x), t]$, and that the map is well-defined on $E$ because when $t=1/2$ (where the two cones meet) the first formula sends $(x\_-, g) = (x, 1/2, g)$ to $[(c(x)g, g), 1/2]$ and the second formula sends the equivalent point $(x\_+, c(x)g) = (x, 1/2, c(x)g)$ to $[(c(x)g, c(x)^{-1} c(x)g], 1/2] = [(c(x)g, g), 1/2]$.


I guess something similar probably works for any topological group $G$, if one uses Milnor's join construction instead of Segal's simplicial model, but I haven't thought through it.


