Maxwell's equations treat the magnetic field as a full and complete continuum, without beginning and without end, not a discrete "lines" that "reconnect".
You are inventing the continuity condition out of thin air.
Maxwell's equations treat the magnetic field as a vector field. Do you know what a vector field is? (a) Every point in space has a unique vector magnitude and (b) if the magnitude is nonzero at any point in space, the field at that point has a unique vector direction.
Do you know what a field line is? It's a particular way of adding up vector field directions from any initial point. Field lines cannot meet or cross in the special case where both (a) the divergence is zero and the magnitude finite, and (b) the field magnitude is nonzero, i.e. where the direction is well defined. If someone told you that field lines can never cross, they were wrong. It is often true that field lines don't cross, because those two conditions ((a) zero divergence and (b) nonzero magnitude) are often met. But when these conditions are not met, crossing field lines do not violate any "continuity" or "smoothness" or anything of vector calculus, and they do not violate Maxwell's Equations.
So, those "cannot cross" conditions are met often but not always. Electric fields, for example, are not divergenceless, so they fail condition (a). Indeed, electric charges always cause electric field lines to cross. (Oh noes!)
Saddle points in electric or magnetic fields, where the field magnitude goes to zero, fail condition (b). There is one point where the magnitude is zero and the field direction is undefined, and at this point the field lines---which, remember, are a particular and well-known sum of vector directions---always cross. There is no "continuity condition" stopping them. There is nothing in Maxwell's Equations stopping them.
All of your insistence to the contrary has been ignoring the actual content of Maxwell's Equations, the actual definition of field lines, and the actual reason (zero divergence, nonzero magnitude) that they *often* seem not to cross. Please stop repeating it.
(Note for people other than MM: "field line crossing" has a well-defined mathematical meaning which we may as well mention. A field line is a particular sort of definite integral over a vector field. Like all definite integrals, it has a different value depending on the starting point. You can say "I want to calculate the field line which begins at point {1,1}", or "... which begins at {10W, 43N}" or whatever. The integral starts by evaluating the field direction at point p. Let's choose an infinitesimal vector ds pointing in this evaluated direction. The integral then evaluates the direction at point (p+ds), picks a new vector ds', and evaluates the direction at (p+ds+ds'). This is unique and reversible; if I had chosen to start at (p+ds+ds') and integrate *backwards*, my first evaluation would have me stepping to (p+ds+ds' - ds') and then to (p+ds+ds' -ds' -ds) I would find myself back at p. The line passing through p, p+ds, p+ds+ds', etc., is called a "field line".
Let's look at the second point, p. How many field lines cross through this point? Well, if I integrate forwards I get to p+ds; if I integrate backwards I get to p-ds. That's a field line crossing through my location. Is it possible to have a second field line, say running from p+dX to p to p-dX? It looks like the answer is "no". If I'm sitting at p, and I evaluate the integral, we think we already know that it will take me to p+ds and not to some other point p+dX, right? So that extra field line, the one passing through p+dX, cannot pass through p ("crossing" the p+ds line) ... if, as we have said, the step away from p necessarily goes to p+ds. BUT what happens if the field magnitude at p is zero? If the magnitude is zero at p, then the direction ds is undefined. If the field is zero at p, then any direction is a valid one to step in in the field line integral. You can step to p+dX, or to p+ds, or to p+dq, or anywhere you want. And some, but not all, of these paths will be reversible. The "backwards" steps from p+dX and p+ds may both point back to p. That means that the line passing through p and p+dX is a valid field line, and the line passing through p and p+ds is another valid field line, and these field lines cross at p.
Notice that the same condition may hold at a point divergence, like a 1/r^2 radial field. Here the field magnitude is not zero (it's infinity in a limit) but the direction is again undefined and ds can be in any direction.
I just want to point out that this is standard vector math; notice that I don't have to say whether the "field" is magnetic, or electric, or fluid flow, or a meaningless math function; notice that I did not mention any extra continuity condition*; that's something MM is making up. Field lines are standard, well-defined properties of all vector fields; field lines may always cross in the way I described. )
(*PPS to people other than MM: I imagine that there is some caveat along the lines of "sufficiently well-behaved" I should state before saying "all vector fields". What, "differentiable"?)