Could the same symmetry be finetuning both the Higgs mass and the inflaton's interactions? The observed Higgs boson mass is at an interesting place in parameter space, placing the standard model electroweak vacuum right at the edge of metastability. Among the proposed explanations of this value is the existence of a "shift symmetry" in the Higgs sector. 
Today we have the possible detection of gravitational waves produced during inflation, of an amplitude suggesting that the inflationary potential was flat right up to near the Planck scale. Various expert commentaries (McAllister, Reece) say that one would expect interactions between the inflaton and the ultraheavy degrees of freedom to appear, and their absence might, once again, imply the existence of a shift symmetry.  
One of the many many approaches to inflation is "Higgs inflation": the Higgs field also serves as the inflaton field, the source of inflation. In discussions of Higgs inflation, I've been told that it's an unlikely model for exactly the same reason as mentioned above - effective field theory ought to break down near the Planck scale, the flatness of the potential should be disturbed by ultraheavy interactions, Higgs inflation would require finetuning. 
But now we may have evidence of such finetuning in inflation! Or perhaps, evidence of a protective symmetry. So my question is, Could the same symmetry be finetuning both the Higgs mass and the inflaton's interactions? Could the same shift symmetry make the Higgs mass critical and protect the inflaton from ultraheavy interactions? 
These questions might be asked first in the context of basic Higgs inflation - only one fundamental scalar - and later in the context of a multi-scalar theory, in which the Higgs is part of a larger scalar sector with a single big potential.
edit: I have found a discussion of shift symmetry in the context of Higgs inflation, but it was written prior to the measurement of the Higgs mass. 
 A: I can now suggest a more specific approach to realizing this concept: conformal Higgs inflation, with a Higgs made critical by a stringy shift symmetry. 
First, see "Higgs inflation at the critical point" and "Higgs inflation still alive". Near the critical point, Higgs inflation predicts too small a value for "r" (r=0.2, approximately, if BICEP2 is to be believed), but right on top of the critical point, sufficiently large values of "r" become possible. Higgs inflation also usually requires ξ, the nonminimal coupling between Higgs field and curvature, to be about 104, but near the critical point, it only needs to be "of order ten" (page 4, ref 1). Ref 2 specifically estimates that ξ of about 7 would lead to r=0.2. 
Meanwhile it has been argued (in "Conformal inflation from the Higgs") that even a ξ as small as -1/6 is viable, because this produces a potential with conformal symmetry, realizing the scenario of "conformal inflation". However, in "Inflation, Symmetry, and B-modes", such models are said to not exhibit a true conformal symmetry at all - instead, they amount to a model of inflation in which there is a scalar field with a shift symmetry. But they also have an unexpected property - they should be subject to perturbative corrections capable of producing a number of observed phenomena, including the large "r". 
As already mentioned, shift symmetries in string theory are capable of producing a critical Higgs - and also a protected inflaton. The inflaton in such scenarios is usually an axion. But if Higgs inflation is phenomenologically viable after all, and especially if "conformal" Higgs inflation would actually be based on a shift symmetry (I acknowledge that these quantitative and conceptual claims are still new and somewhat unexamined)... then it might be time to think about Higgs inflation in the context of string theory.  
