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Made the internal eloops hard so's could be displayed in yellow. Hardness removed for the 2x smooth for large w/f.
Klein bottle
8). Apply Face | Intrude to 'hollow out' the extrusion - it will leave a solid end at the cone apex end.
9). Detail view of 'dead end'
10). Select both objects and apply Object | Weld - this will re-join the (modified) extrusion to the cone.
11). To finish the job off, now need to remove the solid bit at the top of the cone. Select whole object and apply Object | Invert - select both faces.
12). Apply Face | Bridge to form a tunnel.
13). View after cutting thro'.

Finish off as before.

All the above is doing is showing a (very) basic procedure for tackling this 'type' of shape - it shouldn't be too difficult to use this on much more interesting stuff than shown here :)
This page is the result of my involvment with a query (on ODF) about making a Klein bottle in wings.
Purists (and people with far more active grey matter than I) can discuss amongst themselves the complexities of this topic - all I've done is produce some info which might be of use to those that want to make 'something that looks ok'

The first sequence shows a 'bottle' with a 'spout / body' join - the second depicts a modified workflow where the 'spout' passes thro' a hole in the 'body'

As always, what follows is not 'the' way - just an approach that happens to work for the result being sought…

All shortcut keys mentioned are the defaults.
1). Create a cone with 8 sides (click on small square for accessing entry box), select the apex vert and apply Vert | Bevel -> 0.5
2). Select all 8 edges as shown (highlight one, press G)
3). Cut 4 (press 4) all edges and connect (press C)
4). Select / highlight one vert
5). Apply Vert | Bevel -> 0.3 (keep face selected)
6). Press E to change selection to edges.
7). Press 2 to cut all selected edges into 2 equal parts.
8). Select all verts shown (intend to make some quads) and connect (press C)
9). Select all 8 verts around the newly bevelled face (for making circular)
10). Apply Vert | Deform -> Inflate -> 100% (can scale this face if you want)
11). Select the top face of the cone (going to make 'spout')  View -> Z, Ortho, apply Face | Extrude -> Free and drag it out a bit as shown.
12). Apply Face | Rotate -> Free and.
13). Rotate the end face to an eyeball position that looks ok as a joint line with the next extrusion section
14). Situation after a couple of sections have been done.
15). Repeat procedure until only one more section is needed. This extrude / rotate bit is much easier (imo) if you have both commands on hotkeys -
16). Select the 'spout' end face and the face on the cone and apply Face | Bridge to join them together. If you want to tidy up this 'spout' curve, select an edgeloop and use move free / rotate free etc. (all the time staying in View -> Z, of course.)
17). Select the base face (for hollowing out the shape)
18). Apply Face | Intrude ( I used 0.1)
19). Select the edgeloop around the internal spout / body join (highlight one edge, press L)
20). Apply Face | Bevel ->0.25 to form a group of faces (needed to extrude a tube) - I've temporarily removed a bit of cone for clarity purposes.
21). Apply Face | Extrude Region -> Free (View -> Z, ortho again) and use the same technique as before to form this bit of tubing (since this is tube, not solid, you must use extrude region, rather than extrude - to see what happens if you don't, select whole object and smooth it (press S) - try for both, if you want. After finishing the tube, I applied Face | Flatten -> Y on the end faces.
22). Select all (16) faces on the end of the tube and cone.
23). To 'close' base - select (any) one of the newly formed faces on the end of the 'tube' and the corresponding (opposite) one on the inside of the cone. Apply Face | Bridge. Select the edge as shown - the first stage of filling in the rest of the base.
24). Press G to select all the other edges, then F to select all the faces as shown.
25). Apply Face | Dissolve to 'fill in' the rest.
26). Now need to connect some verts to reform some quads from the 2 large faces (one outside - as seen here, one inside the bottle) Connect verts as shown and connect (press C)
27). Now need to repeat te op with the inner face - you can either 'travel inside' to get at the verts required - or as I did here, select whole object, apply Body | Invert and (imo) it's then relatively easy to select them.
28). Apply connect
29). Finshed object after a couple of smooths.
Spout thro' bottle
1). Form face as before (maybe scale out a bit)
2). Apply Face | Inset (I used 30%) keep face selected.
3). Also select base face of cone. (Do NOT select face on top of cone)
4). Apply Face | Intrude -> 0.1
5). Select top face of cone and use the extrude free / rotate stuff to form the solid '?' shape. This time, it will just pass thro' the hole in the side of the cone - no bridging.
6). Select the edgeloop around the top of the cone (where you started the extrusion process from)  and apply Edge | LoopCut to make 2 separate objects.
7). Select end of extrusion.
Spout / body join
Original pix used - left here for ref purposes. Made internal edges hard so's could be displayed in a different colour.