It's far from "a simple answer" and while the non-chiral tops are available finding the Hycaro tippe top of Tokieda is more elusive.
In the paper: "Rattleback: a model of how geometric singularity induces dynamic chirality" by Yoshida, Tokieda and Morrison they explain in the introduction:
"Introduction The chirality in motion, one spin more prominent than the opposite spin, points to an effect that may occur in complex dynamical systems. The minimal model that captures this chiral dynamics is the prototypical rattleback system, PRS $^1$. We shall see that the non dissipative version of PRS admits an odd-dimensional, degenerate Hamiltonian formulation. This is puzzling for two reasons. i) A linearized Hamiltonian system has symmetric spectra, so should be time- reversible; yet chiral dynamics is not. ii) PRS has an extra conserved quantity besides energy, which hitherto has received no intuitive interpretation. The key to the puzzles is a peculiar Lie-algebraic structure behind the Hamiltonian formulation, a so-called Bianchi class B algebra.".
1. H.K. Moffatt, T. Tokieda, Celt reversals: a prototype of chiral dynamics, Proc. Royal Soc. Edinburgh 138A (2008) 361–368.
"Abstract The rattleback is a boat-shaped top with an asymmetric preference in spin. Its dynamics can be described by non-blinearly coupled pitching, rolling, and spinning modes. The chirality, designed into the body as a skewed mass distribution, manifests itself in the quicker transition of +spin → pitch → −spin than that of −spin → roll → +spin. The curious guiding idea of this work is that we can formulate the dynamics as if a symmetric body were moving in a chiral space. By elucidating the duality of matter and space in the Hamiltonian formalism, we attribute asymmetry to space. The rattleback is shown to live in the space dictated by the Bianchi type VI$_{h<−1}$ (belonging to class B) algebra; this particular algebra is used here for the first time in a mechanical example. The class B algebra has a singularity that separates the space (Poisson manifold) into mirror-asymmetric subspaces, breaking the time-reversal symmetry of nearby orbits.".
That paper is referenced in "Towards a Prototype of a Spherical Tippe Top" by Ciocci, Malengier, Langerock, and Grimonprez, also mentioned is this reference:
T. Tokieda, “Private communications,” in Proceedings of the Geometric Mechanics and its Applications (MASIE), Lausanne, Switzerland, July 2004.
Further hypothesis:
The spin direction could match the spin direction of the Earth, adding the difference needed to cause different effects depending upon spin direction. See "rotating coordinate systems" and "Coriolis acceleration" on Wikipedia.
In the paper "On the formation of axial corner vortices during spin-up in a cylinder of square cross-section" by Munro, Hewitt, and Foster, (2015), in the Journal of Fluid Mechanics, 772 . pp. 246-271. ISSN 1469-7645 an acquaintance of Tokieda wrote:
"Once filled, the cylinder was sealed with a transparent rigid lid, which was fitted to completely displace the fluid’s free surface, so that no air pockets were trapped on the lid’s underside.".
This is in reference to a square container filled with fluid, the spin-up speed affects the formation of vortexes in the corners. It's possible for the top to have a square cut from it's center and to be filled with fluid, using one's left or right hand to spin the top provides the necessary difference in speed needed to affect it's operation.
Distantly related, on the manufacturing of these complex shapes:
"Spin-It: Optimizing Moment of Inertia For Spinnable Objects" by Bächer, Bickel, Whiting, and Sorkine-Hornung, on optimizing internal shape:
"In this article, we describe an algorithm to generate designs for spinning objects by optimizing their mass distribution: as input, the user provides a solid 3D model and a desired axis of rotation. Our approach then modifies the interior mass distribution such that the principal directions of the moment of inertia align with the target rotation frame. To create voids inside the model, we represent its volume with an adaptive multiresolution voxelization and optimize the discrete voxel fill values using a continuous, nonlinear formulation. We further optimize for rotational stability by maximizing the dominant principal moment. Our method is well-suited for a variety of 3D printed models, ranging from characters to abstract shapes.
Non-spinning example: Gömböc
"A gömböc or gomboc (Hungarian: [ˈɡømbøt͡s]) is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. Its existence was conjectured by Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by Hungarian scientists Gábor Domokos and Péter Várkonyi. The gömböc shape is not unique; it has countless varieties, most of which are very close to a sphere and all have very strict shape tolerance (about 0.1 mm per 100 mm)).
In the paper "The Turnover of a Tippy Top", Journal of Dynamics and Differential Equations (2015), by H. Ockendon, J. R. Ockendon, and Tokieda, they wrote:
"The turnover of a tippy top is governed by gravity, inertia and, crucially, friction forces. The full model involves 15 differential-algebraic equations with 8 initial conditions and 2 parameters that characterise the top. This paper shows that, in everyday conditions, these equations can be reduced dramatically to reveal the key inequality that must be satisfied for turnover to occur. This inequality involves only the friction coefficient, the geometric and inertial asphericities of the top, the initial tilt and the vertical angular velocity, which remains almost constant.".
In the paper "Dynamics of an axisymmetric body spinning on
a horizontal surface. II. Self-induced jumping", Proceedings of The Royal Society A (2005), by Shimomura, Branicki and Moffat, they wrote:
"... Thus, a spheroid which is spun sufficiently rapidly on a table will, in general, lose contact with the table at some stage during its rising motion. We have considered the free motion only until the first bounce; however, the subsequent behaviour presumably consists of rapidly alternating periods of motion with and without frictional contact with the table. The details of the successive impacts depend on the elastic/plastic properties of both the spheroid and the table. This is clearly a subject for future investigation, both experimental and computational. Irrespective of the details, however, as stated in §1, we may conjecture that the averaged effect (over many successive impacts) will be simply to give an ‘effective’ Coulomb friction parameter me somewhat less than the instantaneous value of $m$ that holds during the periods of continuous contact.".
Along with consideration of the material and it's direction of polish, internal composition, and direction of rotation, the speed of rotation is also an important factor. You can see in the video that he uses two different speeds of rotation versus the direction in which it is spun.
The speed of rotation affects jumping and slippage which is the factor that determines the vector of the spiral course of travel as the peg changes from vertical to horizontal and swings under the center of gravity.
Here is another demonstration of the same Hycaro Tippe Top, I'm not sure it offers new information in the demonstration itself but if you view the video from the beginning it explains chiral dynamics in an easy to understand manner. If there were a slight off-center to the shape it would be chiral.
