For an algebraic number $x$ the property is obvious: Let $a_0+a_1 x+a_2x^2+...+a_nx^n=0$ be the $\mathit{definition}$ of $x$. Divide by $x^n \neq 0$ to get $a_0\left(\frac{1}{x}\right)^n+a_1 \left(\frac{1}{x}\right)^{n-1}+a_2 \left(\frac{1}{x}\right)^{n-2}+...+ a_n =0$ wich has reversed coefficients for $y=\frac{1}{x}$. The accidential minus is possible because $0=-0$. Only RootApproximant is the worst vehicle to show this.

Possibly one could stabilize RootApproximant by forcing it to hold this property because it is going to represent the input as algebraic number.

In[31]:= RootApproximant[SetPrecision[14.6001, #]] & /@ Range[18]
Out[31]= {15, 1/4 (29 + Sqrt[865]), 1/4 (29 + Sqrt[865]), 
 1/2 (15 + Sqrt[201]), 
 1/2 (15 + Sqrt[201]), 73/5, 73/5, 73/5, 73/5, 73/5, 
 Root[63 - 89 #1 - 38 #1^2 + 3 #1^3 &, 3], 
 Root[63 - 89 #1 - 38 #1^2 + 3 #1^3 &, 3], 
 Root[-23 + 37 #1 + 14 #1^2 - 3 #1^4 - 29 #1^5 + 2 #1^6 &, 4], 29171/1998, 
 Root[62 - 8 #1 - 82 #1^2 - 23 #1^3 + 37 #1^4 - 17 #1^5 + #1^6 &, 2], 
 Root[-71 + 44 #1 - 62 #1^2 - 3 #1^3 + 91 #1^4 - 50 #1^5 + 3 #1^6 &, 2], 146001/10000, 146001/10000}

In[32]:= RootApproximant[SetPrecision[1/14.6001, #]] & /@ Range[18]
Out[32]= {1/15, 1/6 (-29 + Sqrt[865]), 1/6 (-29 + Sqrt[865]), 
 1/12 (15 - Sqrt[201]), 
 1/12 (15 - Sqrt[201]), 5/73, 5/73, 5/73, 5/73, 5/73, 
 Root[3 - 38 #1 - 89 #1^2 + 63 #1^3 &, 2], 
 Root[3 - 38 #1 - 89 #1^2 + 63 #1^3 &, 2], 
 Root[-2 + 29 #1 + 3 #1^2 - 14 #1^4 - 37 #1^5 + 23 #1^6 &, 2], 1998/29171, 
 Root[1 - 17 #1 + 37 #1^2 - 23 #1^3 - 82 #1^4 - 8 #1^5 + 62 #1^6 &, 1], 
 Root[2 - 26 #1 - 47 #1^2 - 4 #1^3 + 115 #1^4 + 31 #1^5 + 58 #1^6 &, 1], 10000/146001, 10000/146001}

Forget about it. Change the test a bit more (a PolynomialQ test is missing yet) and drop the degree condition

In[112]:= Clear[mrbQHumble]
mrbQHumble[x_?NumericQ] := 
 Block[{a1 = CoefficientList[First[RootApproximant[x]][y], y], 
   a2 = CoefficientList[First[RootApproximant[1/x]][y], y]},
  If[Length[a1] == Length[a2] && FreeQ[a1, y] && 
    FreeQ[a2, y], (a1 == Reverse[a2]) || (a1 == -Reverse[a2]), 
   Missing[]]
  ]

In[114]:= l0 = RandomReal[{-19, 53}, 100]
Out[114]= {-1.38507, 49.4481, 24.7934, 11.0104, 3.86794, 7.93207, \
48.0579, -9.44529, 48.7661, -14.129, -14.7824, -6.62547, 40.7146, \
-9.87912, -11.0193, 23.8705, -13.5526, 2.14101, 4.57075, -10.0782, \
-3.5094, -5.45227, 30.0554, 39.6618, 3.95229, 13.3368, 43.1726, \
25.5232, 4.30698, -9.89652, 25.0371, 44.8908, 31.7411, -10.7786, \
42.8577, 39.8379, -12.9649, -10.6592, -0.0523882, 24.5372, 52.9216, \
-12.2468, 27.3211, 25.072, -10.8939, 21.1145, -14.224, 52.6378, \
-12.6092, -4.91945, 52.327, 24.0178, 9.16044, 52.1168, 31.1463, \
16.357, 13.9551, 31.295, -8.99637, 43.5836, -14.6001, 34.3883, \
30.7991, 17.0297, -18.3356, -4.35444, 4.17391, 38.1627, 6.66596, \
28.9392, 51.549, -6.69014, 14.7294, -3.10411, 21.2488, 13.6281, \
5.75214, 1.00631, 44.6239, 22.1666, 39.5046, 47.7347, 26.2585, \
52.7089, 50.2851, 13.368, -5.99931, 31.8238, 5.17775, 24.6661, \
9.58021, 19.6149, 7.77517, -6.62582, 16.2579, 46.4658, 15.8527, \
52.8861, -8.88976, 10.9706}

In[115]:= And @@ DeleteMissing[mrbQHumble /@ l0]
During evaluation of In[115]:= First::normal: Nonatomic expression expected at position 1 in First[7335/7289]. >>
During evaluation of In[115]:= First::normal: Nonatomic expression expected at position 1 in First[7289/7335]. >>
Out[115]= False

and the wrong-doer is

In[118]:= RootApproximant[1/l0[[61]]]
Out[118]= Root[-7 - 102 #1 - 2 #1^2 - 72 #1^3 + 4 #1^4 + 65 #1^5 + 44 #1^6 &, 1]

In[119]:= RootApproximant[l0[[61]]]
Out[119]= Root[57 + 118 #1 - #1^2 + 139 #1^3 - 43 #1^4 + 11 #1^5 + #1^6 &, 1]

In[120]:= l0[[61]]
Out[120]= -14.6001