  The Fibonacci sequence modulo $m$, which we denote
$\left(\mathcal{F}_{m,n}\right)_{n=0}^\infty$ where $\mathcal{F}_{m,n}$ is the
Fibonacci number $F_n$ modulo $m$, has been a well-studied object in
mathematics since the seminal paper by D.~D.~Wall in 1960 exploring a myriad of
properties related to the periods of these sequences. Since the time of
Lagrange it has been known that $\left(\mathcal{F}_{m,n}\right)_{n=0}^\infty$
is periodic for each $m$. We examine this sequence when $m=10$, yielding a
sequence of period length 60. In particular, we explore its subsequences
composed of every $r^{\mathrm{th}}$ term of
$\left(\mathcal{F}_{10,n}\right)_{n=0}^\infty$ starting from the term
$\mathcal{F}_{10,k}$ for some $0 \leq k \leq 59$. More precisely we consider
the subsequences $\left(\mathcal{F}_{10,k+rj}\right)_{j=0}^\infty$, which we
show are themselves periodic and whose lengths divide 60. Many intriguing
properties reveal themselves as we alter the $k$ and $r$ values. For example,
for certain $r$ values the corresponding subsequences surprisingly obey the
Fibonacci recurrence relation; that is, any two consecutive subsequence terms
sum to the next term modulo 10. Moreover, for all $r$ values relatively prime
to 60, the subsequence $\left(\mathcal{F}_{10,k+rj}\right)_{j=0}^\infty$
coincides exactly with the original parent sequence
$\left(\mathcal{F}_{10,n}\right)_{n=0}^\infty$ (or a cyclic shift of it)
running either forward or reverse. We demystify this phenomena and explore many
other tantalizing properties of these subsequences.
